Optical profilometry of additional-material deviations in a periodic grating

ABSTRACT

Disclosed is a method and system for measurement of periodic gratings which have deviations which result in more than two materials occurring along at least one line in the periodic direction. A periodic grating is divided into a plurality of hypothetical layers, each hypothetical layer having a normal vector orthogonal to the direction of periodicity, each hypothetical layer having a single material within any line parallel to the normal vector, and at least one of the hypothetical layers having at least three materials along a line in the direction of periodicity. A harmonic expansion of the permittivity ε or inverse permittivity 1/ε is performed along the direction of periodicity for each of the layers including the layer which includes the first, second and third materials. Fourier space electromagnetic equations are then set up in each of the layers using the harmonic expansion of the permittivity ε or inverse permittivity 1/ε, and Fourier components of electric and magnetic fields in each layer. The Fourier space electromagnetic equations are then coupled based on boundary conditions between the layers, and solved to provide the calculated diffraction spectrum.

RELATED DOCUMENTS

The present application is based on Disclosure Document serial number474051, filed May 15, 2000, entitled Optical Profilometry for PeriodicGratings with Three or More Materials per Layer by the same inventors.Therefore, it is requested that the above-specified Disclosure Documentbe retained in the file of the present patent application.

TECHNICAL FIELD

The present invention relates generally to the measurement of periodicsurface profiles using optical techniques such as spectroscopicellipsometry. In particular, the present invention relates to opticalprofilometry of profile deviations of semiconductor fabricationprocesses, more particularly, additional-material deviations in aperiodic grating.

BACKGROUND OF THE INVENTION

There is continual pressure on the semiconductor microchip industry toreduce the dimensions of semiconductor devices. Reduction in the size ofsemiconductor chips has been achieved by continually reducing thedimensions of transistors and other devices implemented on microchiparrays. As the scale of semiconductor devices decreases, control of thecomplete profile of the features is crucial for effective chipoperation. However, limitations in current fabrication technologies makeformation of precise structures difficult. For example, completelyvertical sidewalls and completely horizontal top and bottom surfaces indevice formation are difficult, if not impossible, to achieve. Slopingsidewalls and top and bottom surfaces are common. Additionally, otherartifacts such as “T-topping” (the formation of a “T” shaped profile)and “footing” (the formation of an inverse “T” shaped profile) arecommon in microchip manufacturing. Metrology of such details about theprofile is important in achieving a better understanding of thefabrication technologies. In addition to measuring such features,controlling them is also important in this highly competitivemarketplace. There are thus increasing efforts to develop and refinerun-to-run and real-time fabrication control schemes that includeprofile measurements to reduce process variability.

Optical metrology methods require a periodic structure for analysis.Some semiconductor devices, such as memory arrays, are periodic.However, generally a periodic test structure will be fabricated at aconvenient location on the chip for optical metrology. Optical metrologyof test periodic structures has the potential to provide accurate,high-throughput, non-destructive means of profile metrology usingsuitably modified existing optical metrology tools and off-lineprocessing tools. Two such optical analysis methods include reflectancemetrology and spectroscopic ellipsometry.

In reflectance metrology, an unpolarized or polarized beam of broadbandlight is directed towards a sample, and the reflected light iscollected. The reflectance can either be measured as an absolute value,or relative value when normalized to some reflectance standard. Thereflectance signal is then analyzed to determine the thicknesses and/oroptical constants of the film or films. There are numerous examples ofreflectance metrology. For example, U.S. Pat. No. 5,835,225 given toThakur et.al. teaches the use of reflectance metrology to monitor thethickness and refractive indices of a film.

The use of ellipsometry for the measurement of the thickness of films iswell-known (see, for instance, R. M. A. Azzam and N. M. Bashara,“Ellipsometry and Polarized Light”, North Holland, 1987). When ordinary,i.e., non-polarized, white light is sent through a polarizer, it emergesas linearly polarized light with its electric field vector aligned withan axis of the polarizer. Linearly polarized light can be defined by twovectors, i.e., the vectors parallel and perpendicular to the plane ofincidence. Ellipsometry is based on the change in polarization thatoccurs when a beam of polarized light is reflected from a medium. Thechange in polarization consists of two parts: a phase change and anamplitude change. The change in polarization is different for theportion of the incident radiation with the electric vector oscillatingin the plane of incidence, and the portion of the incident radiationwith the electric vector oscillating perpendicular to the plane ofincidence. Ellipsometry measures the results of these two changes whichare conveniently represented by an angle Δ, which is the change in phaseof the reflected beam ρ from the incident beam; and an angle Ψ, which isdefined as the arctangent of the amplitude ratio of the incident andreflected beam, i.e.,${\rho = {\frac{r_{p}}{r_{s}} = {{\tan (\Psi)}^{j{(\Delta)}}}}},$

where r_(p) is the p-component of the reflectance, and r_(s) is thes-component of the reflectance. The angle of incidence and reflectionare equal, but opposite in sign, to each other and may be chosen forconvenience. Since the reflected beam is fixed in position relative tothe incident beam, ellipsometry is an attractive technique for in-situcontrol of processes which take place in a chamber.

For example, U.S. Pat. No. 5,739,909 by Blayo et. al. teaches a methodfor using spectroscopic ellipsometry to measure linewidths by directingan incident beam of polarized light at a periodic structure. Adiffracted beam is detected and its intensity and polarization aredetermined at one or more wavelengths. This is then compared with eitherpre-computed libraries of signals or to experimental data to extractlinewidth information. While this is a non-destructive test, it does notprovide profile information, but yields only a single number tocharacterize the quality of the fabrication process of the periodicstructure. Another method for characterizing features of a patternedmaterial is disclosed in U.S. Pat. No. 5,607,800 by D. H. Ziger.According to this method, the intensity, but not the phase, ofzeroth-order diffraction is monitored for a number of wavelengths, andcorrelated with features of the patterned material.

In order for these optical methods to be useful for extraction ofdetailed semiconductor profile information, there must be a way totheoretically generate the diffraction spectrum for a periodic grating.The general problem of electromagnetic diffraction from gratings hasbeen addressed in various ways. One such method, referred to as“rigorous coupled-wave analysis” (“RCWA”) has been proposed by Moharamand Gaylord. (See M. G. Moharam and T. K. Gaylord, “RigorousCoupled-Wave Analysis of Planar-Grating Diffraction”, J. Opt. Soc. Am.,vol. 71, 811-818, July 1981; M. G. Moharam, E. B. Grann, D. A. Pommetand T. K. Gaylord, “Formulation for Stable and Efficient Implementationof the Rigorous Coupled-Wave Analysis of Binary Gratings”, J. Opt. Soc.Am. A, vol. 12, 1068-1076, May 1995; and M. G. Moharam, D. A. Pommet, E.B. Grann and T. K. Gaylord, “Stable Implementation of the RigorousCoupled-Wave Analysis for Surface-Relief Dielectric Gratings: EnhancedTransmittance Matrix Approach”, J. Opt. Soc. Am. A, vol. 12, 1077-1086,May 1995.) RCWA is a non-iterative, deterministic technique that uses astate-variable method for determining a numerical solution. Severalsimilar methods have also been proposed in the last decade. (See P.Lalanne and G. M. Morris, “Highly Improved Convergence of theCoupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A, 779-784,1996; L. Li and C. Haggans, “Convergence of the coupled-wave method formetallic lanelar diffraction gratings”, J. Opt. Soc. Am. A, 1184-1189,June, 1993; G. Granet and B. Guizal, “Efficient Implementation of theCoupled-Wave Method for Metallic Lamelar Gratings in TM Polarization”,J. Opt. Soc. Am. A, 1019-1023, May, 1996; U.S. Pat. No. 5,164,790 byMcNeil, et al; U.S. Pat. No. 5,867,276 by McNeil, et al; U.S. Pat. No.5,963,329 by Conrad, et al; and U.S. Pat. No. 5,739,909 by Blayo et al.)

Generally, an RCWA computation consists of four steps:

The grating is divided into a number of thin, planar layers, and thesection of the ridge within each layer is approximated by a rectangularslab.

Within the grating, Fourier expansions of the electric field, magneticfield, and permittivity leads to a system of differential equations foreach layer and each harmonic order.

Boundary conditions are applied for the electric and magnetic fields atthe layer boundaries to provide a system of equations.

Solution of the system of equations provides the diffracted reflectivityfrom the grating for each harmonic order.

The accuracy of the computation and the time required for thecomputation depend on the number of layers into which the grating isdivided and the number of orders used in the Fourier expansion.

The diffracted reflectivity information which results from an RCWAcomputation can be used to determine the details of the profile of asemiconductor device. Generally, reflectivities for a range of differentpossible profiles of a given semiconductor device are numericallycalculated using RCWA and stored in a database library. Then, the actualdiffracted reflectivity of the given device is measured as disclosed,for example, in co-pending U.S. patent application Ser. No. 09/764,780for Caching of Intra-Layer Calculations for Rapid Rigorous Coupled-WaveAnalyses filed Jan. 25, 2000 by the present inventors which is herebyincorporated in its entirety into the present specification, or X. Niu,N. Jakatdar, J. Bao and C. J. Spanos, “Specular SpectroscopicScatterometry” IEEE Trans. on Semiconductor Manuf., vol. 14, no. 2, May2001. The reflected phase and magnitude signals obtained, in the case ofellipsometry, and relative reflectance, in the case of reflectometry,are then compared to the library of profile-spectra pairs generatedstored in the library. A phase and/or amplitude measurement will bereferred to in the present specification as the “diffractedreflectivity.” The matching algorithms that can be used for this purposerange from simple least squares approach, to a neural network approachthat associates features of the signal with the profile through anon-linear relationship, to a principal component based regressionscheme. Explanations of each of these methods is explained in numeroustext books on these topics such as Chapter 14 of “MathematicalStatistics and Data Analysis” by John Rice, Duxbury Press and Chapter 4of “Neural Networks for Pattern Recognition” by Christopher Bishop,Oxford University Press. The profile associated with the RCWA-generateddiffracted reflectivity that most closely matches the measureddiffracted reflectivity is determined to be the profile of the measuredsemiconductor device.

In semiconductor manufacturing, a number of processes may be used toproduce a periodic structure having two materials in the periodicdirection. In the present specification the “nominal” number ofmaterials occurring in the periodic direction is considered to be themaximum number of materials that lie along any of the lines which passthrough the periodic structure in the direction of the periodicity.Accordingly, structures having a nominal two materials in the periodicdirection have at least one line along the direction of periodicitypassing through two materials, and no lines along the direction ofperiodicity passing through more than two materials. Additionally, itshould be noted that when specifying the nominal number of materialsoccurring along a periodic direction of a structure in the presentspecification, the gas, gases or vacuum in gaps between solid materialsis considered to be one of the materials. For instance, it is notnecessary that both materials occurring in the periodic direction of anominal two material periodic structure be solids.

An example of a structure 100 with two materials in a layer is shown inthe cross-sectional view of FIG. 1A, which shows two periods of length Dof a periodic portion of the structure 100. The structure 100 consistsof a substrate 105, with a thin film 110 deposited thereon, and aperiodic structure on the film 110 which consists of a series of ridges121 and grooves 122. In exemplary structure 100, each ridge 121 has alower portion 131, a middle portion 132 and an upper portion 133. Itshould be noted that according to the terminology of the presentinvention, the lower, middle and upper portions 131-133 are not‘layers.’ In exemplary structure 100 of FIG. 1A, the lower, middle andupper portions 131-133 are each composed of a different material. Thedirection of periodicity is horizontal on the page of FIG. 1A, and itcan be seen that a line parallel to the direction of periodicity maypass through at most two different materials. For instance, a horizontalline passing through the middle portion 132 of one of the ridgestructures 121, passes through the middle portion 132 of all of theridge structures 121, and also passes through the atmospheric material122. That is, there are two materials in that region. (It should benoted that a line which is vertical on the page of FIG. 1A can passthrough more than two materials, such as a line passing through thelower, middle and upper portions 131-133 of a ridge structure, the thinfilm 110, and the substrate 105, but according to the terminology of thepresent specification this structure 100 is not considered to have anominal three or more materials in the periodic direction.)

A close-up cross-sectional view of a ridge structure 121 is shown inFIG. 1B with the structure being sectioned into what are termed‘harmonic expansion layers’ or simply ‘layers’ in the presentspecification. In particular, the upper portion 133 is sectioned intofive harmonic expansion layers 133.1 through 133.5, the middle portion132 is sectioned into nine harmonic expansion layers 132.1 through132.9, the lower portion 131 is sectioned into six harmonic expansionlayers 131.1 through 131.6, and five harmonic expansion layers 110.1through 110.5 of the thin film 110 are shown. All layer boundaries arehorizontal planes, and it should be understood that harmonic expansionlayers 133.1-133.5, 132.1-132.9, 131.1-131.6 and 110.1-110.5 may havediffering thicknesses. For clarity of depiction, the harmonic expansionlayers 133.1-133.5, 132.1-132.9, 131.1-131.6 and 110.1-110.5 are notshown to extend into the atmospheric material, although they areconsidered to do so. As can be seen in FIG. 1A, a structure having twomaterials occurring in a periodic direction will necessarily have twomaterials in an harmonic expansion layer.

With respect to semiconductors having a periodic structure with anominal two materials in periodic direction, it is often the case thatthe widths of the solid structures in the periodic direction isimportant to proper operation of the device being produced. For example,the width of a structure (such as a transistor gate) can determine howquickly or slowly a device will operate. Similarly, the width of aconductor can determine the resistance of the conductor, or the width ofa gap between two conductors can determine the amount of currentleakage. Furthermore, the geometry of a structure in the periodicdirection can also impact the geometry of successive layers of the chip.

Because the characteristic dimension of a structure in a directionorthogonal to the normal vector of the substrate generally has the mostimpact on the operation of a device and the fabrication of thecharacteristic dimension in successive layers of the chip, thatdimension is referred to as the “critical” dimension. Because of theimportance of critical dimension, it is common to use both the RCWAtechniques discussed above and various other types of microscopy (suchas critical-dimension scanning electron microscopy, cross-sectionalscanning electron microscopy, atomic force microscopy, and focused ionbeam measurement) to measure critical dimensions. While these techniquescan generally adequately measure critical dimensions of structureshaving a single solid material along a line in the periodic direction,none of these techniques can make accurate measurements of criticaldimensions of multiple material components of structures when more thana single solid material occurs in the periodic direction. In particular,such techniques generally cannot make accurate measurements of materialshaving more than two materials in a periodic direction.

However, a process which is intended to produce a structure with onlytwo materials per layer may have deviations which result in more thantwo materials in a layer. For example, in FIG. 2A a semiconductor device810 is shown in which troughs 812 have been etched in a vertical portion814, such as a series of ridges 815. Such a process nominally produces astructure having two materials along each line in the periodicdirection: the solid material of the ridges 815 and the atmosphericmaterial in the troughs 812. However, as shown in FIG. 2B, whichillustrates a common manufacturing defect on semiconductor device 810,when etching the troughs 812, a thin polymer layer 818 can remain coatedon the side and bottom walls of the troughs 812.

Therefore, device 810 has three materials along the line 820 in theperiodic direction: the material of ridges 815, the material of polymer818, and the atmospheric gas in trough 812. And, as noted above,techniques discussed above which can measure critical dimensions ofperiodic structures having a nominal two materials in the periodicdirection cannot be used to accurately measure the dimensions ofmultiple solid materials within structures having more than twomaterials in the periodic direction. Specifically, techniques ordinarilyused to measure the width of the ridges 815 will not yield an accuratemeasurement result when polymer 818 is present. This is because suchtechniques generally cannot distinguish between the material of ridges815 and the material of polymer 818.

A second example of a structure which is intended to nominally have onlytwo materials in the periodic direction but which, due toadditional-material deviations, has more than two materials in theperiodic direction can occur in performing chemical mechanical polishing(“CMP”), as is shown in FIG. 3A. FIG. 3A shows a semiconductor device700 having a substrate 710 with a nitride layer 714 formed thereon.Troughs 712 are etched in the substrate 710 and nitride layer 714.Silicon dioxide plugs 716 are then placed in troughs 712. This resultsin a periodic structure which has either one or two materials in theperiodic direction. In particular, the substrate material 710 and thematerial of the silicon dioxide plugs 716 fall along line 722; thematerial of nitride layer 714 and the material of the silicon dioxideplugs 716 fall along line 718; and the material of the substrate 710falls along line 724.

After the silicon dioxide plugs 716 have been formed, such a device 700would typically be further processed using a technique referred to as“shallow trench isolation CMP”. This technique is intended to smooth thetop face of the device so that the top of nitride layer 714 and the topof the silicon dioxide plugs 716 both come to the same level, shown byline 720. However, because silicon dioxide is softer than nitride,silicon dioxide plugs 716 will erode further than the nitride layer 714.This results in portions of silicon dioxide plugs 716 dipping below thetop surface of the nitride layer, and is known as “dishing” of silicondioxide plugs 716. And, as shown in FIG. 3A along line 718, near the topof the nitride layer 714, device 700 can has three materials occurringin the periodic direction: nitride, silicon dioxide and the atmosphericmaterial in those regions where the dishing has resulted in the topsurface 717 of the silicon dioxide plugs 716 being below the level ofline 718.

This type of deviation is referred to in the present specification as a“transverse” deviation because it is transverse to the periodicdirection of the structure and is transverse to what would generally bethe direction along which the critical dimension is measured. That is,the deviation occurs in the direction normal to the face of device 700(in a vertical direction in FIG. 3), rather than along the periodicdirection. In contrast, the semiconductor manufacturing industrygenerally focuses on deviations in the critical dimension, such asT-topping discussed earlier. Accordingly, the idea of measuring theextent of any dishing occurring in a semiconductor manufacturing processhas not generally arisen in the semiconductor fabrication industry sincetransverse deviations have not been considered to have substantialeffects on the operation of devices or the fabrication of subsequentlayers.

However, it is here predicted that with continuing technologicalinnovations allowing the size of semiconductor devices to steadilyshrink, the functioning of semiconductor devices will becomeincreasingly dependent on precise fabrication control and metrologyalong the transverse direction, and precise fabrication and control ofadditional-material deviations. Furthermore, recently developed deviceshave been designed with their critical dimension (i.e., the dimensionhaving the greatest effect on the operation of the device) along thenormal to the substrate, i.e., along the direction that the presentspecification has previously referred to as the transverse direction.Therefore, it is here predicted that future generations of semiconductorsystems will both have devices with their critical dimension parallel tothe substrate, and devices with their critical dimension perpendicularto the substrate.

SUMMARY OF THE INVENTION

A method and system in accordance with the present invention allowsmeasurement of semiconductor fabrication methods which ideally have onlytwo materials along a line in a periodic direction, but which havedeviations which result in more than two materials occurring along aline in a periodic direction.

A method for metrology of additional-material structural deviations of anominal periodic structure by comparison of a measured diffractionspectrum from a target periodic structure with a calculated diffractionspectrum from a hypothetical deviated periodic structure, where thehypothetical deviated periodic structure is defined by applying theadditional-material structural deviations to said nominal periodicstructure. The hypothetical deviated periodic structure has a directionof periodicity x, a direction of essentially-infinite extension y whichis orthogonal to the x direction, and a normal direction z which isorthogonal to both the x and y directions. A plurality of layers aredefined parallel to an x-y plane. An x-z plane cross-section of theperiodic structure is sectioned into a plurality of stacked rectangularsections such that only two materials from the nominal periodicstructure are within each of the plurality of layers and at least threematerials are within at least one of the plurality of layers in thehypothetical deviated periodic structure. A harmonic expansion of afunction of the permittivity ε is performed along the direction ofperiodicity x for each of the layers, including the layer or layers inthe hypothetical deviated periodic structure which include(s) at leastthree materials. Fourier space electromagnetic equations are then set upin each of the layers using the harmonic expansion of the function ofthe permittivity ε for each of the layers and Fourier components ofelectric and magnetic fields in each layer. The Fourier spaceelectromagnetic equations are then coupled based on boundary conditionsbetween the layers, and solved to provide the calculated diffractionspectrum.

In a second aspect of the present invention, generation of thediffracted reflectivity of a periodic grating to determine values ofstructural properties of the periodic grating includes dividing theperiodic grating into a plurality of hypothetical layers at least one ofwhich is formed across at least first, second and third materials in theperiodic grating. Each hypothetical layer has its normal vectororthogonal to the direction of periodicity, and each hypothetical layerhas one of a plurality of possible combinations of hypothetical valuesof properties for that hypothetical layer. Sets of hypothetical layerdata are then generated. Each set of hypothetical layer data correspondsto a separate one of the plurality of hypothetical layers. The generatedsets of hypothetical layer data are processed to generate the diffractedreflectivity that would occur by reflecting electromagnetic radiationoff the periodic grating.

Preferably, each hypothetical layer is subdivided into a plurality ofslab regions with each slab region corresponding to a separate materialwithin the hypothetical layer. Also, preferably, generating sets ofhypothetical layer data includes expanding the real space permittivityor the real space inverse permittivity of the hypothetical layers in aone-dimensional Fourier transformation along the direction ofperiodicity of the periodic grating. Preferably, the Fourier transformis formulated as a sum over boundaries between materials in each layer.

In a third aspect of the present invention, a method of generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction for use in an opticalprofilometry formalism for determining a diffracted reflectivity of thetarget periodic grating includes dividing the target periodic gratinginto a plurality of hypothetical layers. At least one of thehypothetical layers is formed across each of at least a first, secondand third material occurring along a line parallel to a direction ofperiodicity of the target periodic grating. At least one of theplurality of hypothetical layers is subdivided into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries.Each of the plurality of hypothetical boundaries corresponds to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials. Apermittivity function is determined for each of the plurality ofhypothetical layers. Then, a one-dimensional Fourier expansion of thepermittivity function of each hypothetical layer is completed along thedirection of periodicity of the target periodic grating by summing theFourier components over the plurality of hypothetical boundaries toprovide harmonic components of the at least one permittivity function. Apermittivity harmonics matrix is then defined including the harmoniccomponents of the Fourier expansion of the permittivity function.

A system of the present invention includes a microprocessor configuredto perform the steps of the methods discussed above. Additionally, acomputer readable storage medium in accordance with the presentinvention contains computer executable code for instructing a computerto operate to complete the steps of the methods discussed above.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a cross-sectional view showing a periodic structure wherethere is a maximum of two materials along any line in the periodicdirection.

FIG. 1B is a close-up cross-sectional view of one of the ridges of FIG.1A with the ridge being sectioned into layers.

FIG. 2A is a cross-sectional view of a semiconductor device havingetched troughs and in which at most two materials occur along any linein the periodic direction.

FIG. 2B is a cross-sectional view of a the semiconductor device shown inFIG. 2A in which a residual polymer layer coats the etched troughsresulting in more than two materials unintentionally occurring along anyline in the periodic direction.

FIG. 3A is a cross-sectional view of a semiconductor device whichincludes transverse deviations resulting in more than two materialsoccurring a line in the periodic direction.

FIG. 3B is a cross-sectional view of the semiconductor device of FIG. 3Awithout the transverse deviations which result in more than twomaterials occurring along a line in the periodic direction.

FIG. 4 shows a section of a diffraction grating labeled with variablesused in a mathematical analysis in accordance with the presentinvention.

FIG. 5 is a cross-sectional view of the semiconductor shown in FIG. 3Asectioned into harmonic expansion layers and discretized intorectangular slabs in accordance with the present invention.

FIG. 6 shows a process flow of a TE-polarization rigorous coupled-waveanalysis in accordance with the present invention.

FIG. 7 shows a process flow for a TM-polarization rigorous coupled-waveanalysis in accordance with the present invention.

FIG. 8 is a cross-sectional view of a drain region of a semiconductordevice illustrating formation of spacers in a lightly doped drainstructure and having more than two materials along a line in theperiodic direction.

FIG. 9 is a flow chart illustrating a method of generating an expressionof the permittivity of a target periodic grating having more than twomaterials in a periodic direction in accordance with the presentinvention.

FIG. 10 illustrates a computer system for implementation of thecomputation portions of the present invention.

DETAILED DESCRIPTION

The present invention relates to metrology of additional-materialdeviations and deviations in a direction transverse to the criticaldimension using a diffraction calculation technique. A system and methodin accordance with the present invention can be used for the measurementof one-dimensionally periodic surface profiles, particularly where thesurface profile has three or more materials along at least one line inthe periodic direction.

FIG. 4 is a diagram of a section of a periodic grating 600. The sectionof the grating 600 which is depicted includes three ridges 621 which areshown as having a triangular cross-section. It should be noted that themethod of the present invention is applicable to cases where the ridgeshave shapes which are considerably more complex, and even to cases wherethe categories of “ridges” and “troughs” may be ill-defined. Accordingto the lexography of the present specification, the term “ridge” will beused for one period of a periodic structure on a substrate. Each ridge621 of FIG. 4 is considered to extend infinitely in the +y and −ydirections, and an infinite, regularly-spaced series of such ridges 621are considered to extend in the +x and −x directions. The ridges 621 areatop a deposited film 610, and the film 610 is atop a substrate 605which is considered to extend semi-infinitely in the +z direction. Thenormal vector {right arrow over (n)} to the grating is in the −zdirection.

FIG. 4 illustrates the variables associated with a mathematical analysisof a diffraction grating according to the present invention. Inparticular:

θ is the angle between the Poynting vector of the incidentelectromagnetic radiation 631 and the normal vector {right arrow over(n)} of the grating 600. The Poynting vector and the normal vector{right arrow over (n)} define the plane of incidence 640.

φ is the azimuthal angle of the incident electromagnetic radiation 631,i.e., the angle between the direction of periodicity of the grating,which in FIG. 4 is along the x axis, and the plane of incidence 640.(For ease of presentation, in the mathematical analysis of the presentspecification the azimuthal angle φ is set to zero.)

ψ is the angle between the electric-field vector {right arrow over (E)}of the incident electromagnetic radiation 631 and the plane of incidence640, i.e., between the electric field vector {right arrow over (E)} andits projection {right arrow over (E)}′ on the plane of incidence 640.When φ=0 and the incident electromagnetic radiation 631 is polarized sothat ψ=π/2, the electric-field vector {right arrow over (E)} isperpendicular to the plane of incidence 640 and the magnetic-fieldvector {right arrow over (H)} lies in the plane of incidence 640, andthis is referred to as the TE polarization. When φ=0 and the incidentelectromagnetic radiation 631 is polarized so that ψ=0, themagnetic-field vector {right arrow over (H)} is perpendicular to theplane of incidence 640 and the electric-field vector {right arrow over(E)} lies in the plane of incidence 640, and this is referred to as theTM polarization. Any planar polarization is a combination of in-phase TEand TM polarizations. The method of the present invention describedbelow can be applied to any polarization which is a superposition of TEand TM polarizations by computing the diffraction of the TE and TMcomponents separately and summing them. Furthermore, although the‘off-axis’ φ≠0 case is more complex because it cannot be separated intoTE and TM components, the present invention is applicable to off-axisincident radiation as well.

λ is the wavelength of the incident electromagnetic radiation 631.

FIG. 5 illustrates division of the periodic structure of FIG. 3A into aplurality of expansion layers to allow a mathematical analysis of thediffraction grating in accordance with the present invention. In thecoordinate system 211 shown in FIG. 5, the periodic direction is the xdirection, the transverse direction is the z direction, and the ydirection is a direction of essentially infinite extension orthogonal tothe x direction and z direction normal to the page.

As described above in reference to FIG. 3A, the periodic structure 700includes a substrate 710 with a nitride layer 714 formed thereon.Troughs 712 are etched in a periodic manner in the substrate 710 andnitride layer 714. Silicon dioxide plugs 716 are then placed in troughs712. As explained in the Background section above, because silicondioxide is softer than nitride, when a CMP process is applied to thesemiconductor device, silicon dioxide plugs 216 will erode further thannitride layer 214. This results in portions of silicon dioxide plugs 216dipping below the top surface of the nitride layer 714 and creating atransverse deviation. In particular, near the top surface of the nitridelayer 714, the semiconductor device has three materials occurring alonga line in the periodic direction: nitride, silicon dioxide andatmospheric gas.

FIG. 5 illustrates the variables associated with a mathematicaldescription of the dimensions of exemplary grating 700 according to thepresent invention. The nominal profile of FIG. 5 (i.e., the profile thatwould occur in this case if there was no dishing) has one or twomaterials per layer: the material of the substrate 710 and the silicondioxide of the plugs 716 in layers 225.4 and 225.5; the material of thesubstrate 710 in layers 225.6 and 225.7; the material of the substrate710 and the nitride of the nitride layer 714 in layers 225.1, 225.2 and225.3; and the material of the substrate 710 in layer 225.0. The dishingof the plugs 716 is considered to be an additional-material deviationprior to discretization. Also, atmospheric slabs 244.1 c and 244.2 c areconsidered to be additional-material deviations of the discretizedprofile. Accordingly, in layers 225.1 and 225.2 there are threematerials: the atmospheric material in slabs 244.1 c and 244.2 c, thenitride in slabs 238.1 and 238.2, and the silicon dioxide in slabs 244.1a , 244.1 b, 244.1 b, and 244.2 b.

FIG. 9 is a flow chart illustrating a method of generating a diffractedreflectivity of a target periodic grating, such as grating 700 of FIG.5, having additional material deviations resulting in a grating with 2or more materials occurring along a periodic direction. Specifically,FIG. 9 illustrates a method in accordance with the present invention forexpressing the permittivity of a target periodic grating having 2 ormore materials occurring along a periodic direction, this expression isreferred to herein generally as hypothetical layer data. As expressed,this hypothetical layer data can be used to generate a theoretical orsimulated diffracted reflectivity of the target periodic grating. FIGS.6 and 7 illustrate the details of a method in accordance with thepresent invention of determining a theoretical diffracted reflectivityof a target periodic grating using the hypothetical layer data. FIG. 6illustrates this process flow for TE-polarization rigorous coupled-waveanalysis in accordance with the present invention and FIG. 7 illustratesthis process flow for a TM-polarization rigorous coupled-wave analysisin accordance with the present invention.

Referring first to FIG. 9, in step 10, the target periodic grating 700(shown in FIG. 5) is divided into hypothetical harmonic expansionlayers. Referring again to FIG. 5, L+1 is the number of the harmonicexpansion layers into which the system is divided. Harmonic expansionlayers 0 and L are considered to be semi-infinite layers. Harmonicexpansion layer 0 is an “atmospheric” layer 701, such as vacuum or air,which typically has a refractive index n_(o) near unity. Harmonicexpansion layer L is a “substrate” layer 710, which is typically siliconor germanium in semiconductor applications. In the case of the exemplarygrating 700, there are eight harmonic expansion layers, with theatmospheric layer 701 above grating 700 being the zeroth harmonicexpansion layer 225.0; the first and second harmonic expansion layers225.1 and 225.2, respectively, containing a top portion of the nitridelayer 714, the dished portion of the silicon dioxide plugs 712, and theatmospheric material; the third harmonic expansion layer 225.3containing the bottom portion of the nitride layer 714, and a middleportion of the silicon dioxide plugs 712; the fourth and fifth harmonicexpansion layers 225.4 and 225.5, respectively, containing the materialof the substrate 710 and the bottom portion of the silicon dioxide plugs716; and the sixth and seventh harmonic expansion layers 225.6 and 225.7containing only the material of the substrate 710. (Generically orcollectively, the harmonic expansion layers are assigned referencenumeral 225, and, depending on context, “harmonic expansion layers 225”may be considered to include the atmospheric layer 201 and/or thesubstrate 205.) As shown in FIG. 5, the harmonic expansion layers areformed parallel to the direction of periodicity of the grating 700. Itis also considered, however, that the layers form an angle with thedirection of periodicity of the grating being measured.

Referring again to FIG. 9, after dividing grating 700 into thehypothetical harmonic expansion layers as described above, in step 12,the hypothetical harmonic expansion layers are further divided intoslabs defined by the intersections of the harmonic expansion layers withthe materials forming the periodic grating. As shown in FIG. 5, thesection of each material within each intermediate harmonic expansionlayers 225.1 through 225.(L−1) is approximated by a planar slabs ofrectangular cross-section 238.1, 238.2, 238.3, 248.1, 248.2, 244.1 a,244.1 c, 244.1 b, 244.2 a, 244.2 c, 244.2 b, 244.3, 244.4, 244.5, 250.1,and 250.2. The top and bottom surfaces of each slab are located at theboundaries between harmonic expansion layers. The side surfaces of eachslab are vertical and are located at the boundary between materials whenthat boundary is vertical, or across the boundary between materials whenthat boundary is not vertical. For instance, as shown in FIG. 5, slab244.1 a has its left sidewall at the boundary between the nitride layer714 and the plug 716. The right wall of slab 244.1 a crosses theboundary between the plug 716 and the atmospheric material at a pointpartway between leftmost edge of that boundary and rightmost edge ofthat boundary. Similarly, slab 244.1 b has its right sidewall at theboundary between the nitride layer 714 and the plug 716, and the leftwall of slab 244.1 b crosses the boundary between the plug 716 and theatmospheric material at a point partway between leftmost edge of thatboundary and rightmost edge of that boundary. Between slabs 244.1 a and244.1 b is slab 244.1 c. The left sidewall of slab 244.1 c is coincidentwith the right sidewall of slab 244.1 a, and the right sidewall of slab244.1 c is coincident with the left sidewall of slab 244.1 b. Clearly,any geometry of exemplary grating 700 with a cross-section which doesnot consist solely of vertical and horizontal borders can be betterapproximated using a greater number of harmonic expansion layers 225.For example, in practice, the portion of exemplary grating 700 inharmonic expansion layers 225.1 and 225.2 might be divided into a largernumber of harmonic expansion layers so that the vertical sidewallsacross the curved dishing surface 717 would be better approximated.However, for the sake of clarity, this region of exemplary grating isdivided into only two harmonic expansion layers 225.1 and 225.2.

Other parameters shown in FIG. 5 are as follows:

D is the periodicity length or pitch, i.e., the length betweenequivalent points on pairs of adjacent ridges.

x^((l)) _(k) is the x coordinate of the starting border of the kthmaterial in the lth layer, and x^((l)) _(k−1) is the x coordinate of theending border of the kth material in the lth layer, so that x^((l))_(k)−x⁽¹⁾ _(k−1) is the width of the kth slab in the lth layer. Forexample, as shown in FIG. 5, x^((l)) ₂−x^((l)) _(l) is the width of thenitride slab 238.1.

t_(l), is the thickness of the lth layer 225.1 for 1<l<(L−1). Thethicknesses t_(l) of the layers 225 are chosen so that (after thediscretization of the profile) every vertical line segment within eachlayer 225 passes through only a single material. For instance, prior todiscretization a vertical line in the region of slab 244.1 a would passthrough the boundary 717 between the atmospheric material and thesilicon dioxide. However, upon discretization, where that region isreplaced by a slab 244.1 a of silicon dioxide, any vertical line in thatregion only passes through the silicon dioxide

n_(k) is the index of refraction of the kth material in grating 700.

In determining the diffraction generated by grating 700, as discussed indetail below, a Fourier space version of Maxwell's equations is used.Referring again to FIG. 9, to generate these equations, in step 14,hypothetical layer data is generated by completing a harmonic expansionof a function of the permittivities of the materials in the targetperiodic grating. In Step 14 a of FIG. 9 (step 310 of FIG. 6 for TEpolarization and step 410 of FIG. 7 for TM polarization) thepermittivities ε_(l)(x) for each layer l are determined or acquired asis known by those skilled in the art and disclosed, for example, in U.S.patent application Ser. No. 09/728,146 filed Nov. 28, 2000 entitledProfiler Business Model, by the present inventors which is incorporatedherein by reference in its entirety. A one-dimensional Fouriertransformation of the permittivity ε_(l)(x) or the inverse permittivityπ_(l)(x)=1/ε_(l)(x) of each layer l is performed in step 14 b or FIG. 9(step 312 of FIG. 6 and step 412 of FIG. 7) along the direction ofperiodicity, x, of the periodic grating 700 to provide the harmoniccomponents of the permittivity ε_(l,i) or the harmonic components of theinverse permittivity π_(l,i), where i is the order of the harmoniccomponent. In particular, the real-space permittivity ε_(l)(x) of thelth layer is related to the permittivity harmonics ε_(l,i) of the lthlayer by $\begin{matrix}{{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}.}}}} & \text{(1.1.1)}\end{matrix}$

Therefore, via the inverse transform, $\begin{matrix}{{ɛ_{0} = {{\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}x_{k - 1}}} - \frac{x_{k}}{D}}},} & \text{(1.1.2)}\end{matrix}$

and for i not equal to zero, $\begin{matrix}{ɛ_{l,i} = {\sum\limits_{k = 1}^{r}\quad {{\frac{n_{k}^{2}}{- {{j}2\pi}}\left\lbrack {\left( {{\cos \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\sin \left( {\frac{2i\quad \pi}{D}x_{k - 1}} \right)}} \right)}} \right\rbrack}.}}} & \text{(1.1.3)}\end{matrix}$

where the sum is over the number r of borders and n_(k) is the index ofrefraction of the material between the kth and the (k−1)th border and jis the imaginary number defined as the square root of −1. Similarly, theinverse of the permittivity, π_(l,i), of the lth layer is related to theinverse-permittivity harmonics π_(l,i) of the lth layer by$\begin{matrix}{{\pi_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}{{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}.}}}} & \text{(1.1.4)}\end{matrix}$

Therefore, via the inverse transform, $\begin{matrix}{{\pi_{0} = {{\sum\limits_{k = 1}^{r}\quad {n_{k}^{- 2}x_{k - 1}}} - \frac{x_{k}}{D}}},} & \text{(1.1.5)}\end{matrix}$

and for i not equal to zero, $\begin{matrix}{\pi_{l,i} = {\sum\limits_{k = 1}^{r}\quad {{\frac{n_{k}^{- 2}}{- {{j}2\pi}}\left\lbrack {\left( {{\cos \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\sin \left( {\frac{2i\quad \pi}{D}x_{k - 1}} \right)}} \right)}} \right\rbrack}.}}} & \text{(1.1.6)}\end{matrix}$

where the sum is over the number r of borders and n_(k) is the index ofrefraction of the material between the kth and the (k−1)th border and jis the imaginary number defined as the square root of −1. It isimportant to note that equations for the harmonic components of thepermittivity ε or inverse permittivity π provided by the prior art areformulated as a sum over materials, and are only directed towardsituations where each harmonic expansion layer has only one or twomaterials. In contrast, equations (1.1.2) and (1.1.3) and equations(1.1.5) and (1.1.6) are formulated as sums over the boundaries betweendifferent materials occurring in the periodic direction, and can handlegeometries with any number of materials in a harmonic expansion layer.

As such, the system and method of the present invention is not onlyapplicable to the semiconductor device 700 shown in FIG. 5, but also todevices exhibiting other types of deviations, such as semiconductordevice 810 shown in FIG. 2B which includes a polymer layer.

Additionally, the system and method of the present invention could beused to measure structural dimensions of a periodic grating which bydesign have three or more materials occurring along a line in a periodicdirection. One example of such a device is a field effect transistor 740shown in FIG. 8 having a source 742 a, a drain 742 b, and a gate 746.The gate 746 is placed on top of a insulating oxide barrier layer 744which coats the substrate 742. A voltage applied to the gate 746produces an electric field in the region between the source 742 a andthe drain 742 b which strongly affects current flow between the source742 a and the drain 742 b. A top left spacer 748 at is formed on theleft side of the gate 746 on top of the barrier layer 744, a top rightspacer 748 bt is formed on the right side of the gate 746 on top of thebarrier layer 744, a bottom left spacer 748 ab is formed on the leftside of the gate 746 below the barrier layer 744 and to the right of thesource 742 a, and a bottom right spacer 748 bb is formed on the rightside of the gate 746 below the barrier layer 744 and to the left of thedrain 742 b. The lower spacers 748 ab and 748 bb reduce the magnitude ofelectric field gradients in the regions near the source 742 a and 742 band below the barrier layer 744, thereby preventing the amount ofcurrent which “jumps” through the barrier layer 744. The sizes andshapes of spacers 748 impacts the operation of device 740. If thespacers 748 at, 748 ab, 748 bt and 748 bb are too large, the operationof device 740 can be too slow. However, if the lower spacers 748 ab and748 bb are too small, current leakage through the barrier layer 744 canoccur, resulting in defective operation. Accordingly, it can beimportant to monitor the width of spacers 748. It should be noted thatthree materials lie along line 750: the material in the atmosphere 741,the material of the top spacers 748 at and 748 bt, and the material ofthe gate 746. Similarly, four materials lie along line 760: the materialof the substrate 742, the material of the lower spacers 748 ab and 748bb, the material of the source 742 a, and the material of the drain 742b.

The method disclosed herein of describing a periodic grating such as asemiconductor device by dividing the grating into layers as discussedabove and shown in FIG. 5, and expanding the permittivity of a periodicgrating such as grating 700 as shown in equations (1.1.1) through(1.1.3), or expanding the inverse permittivity as shown in equations(1.1.4) through (1.1.6), can be used with any optical profilometryformalism for determining a diffracted reflectivity that uses a Fouriertransform of the permittivity or inverse permittivity.

Referring again to FIG. 9, in step 16, the sets of hypothetical layerdata generated as described above are processed to generate thediffracted reflectivity. This step involves three general sub-steps:First, in sub-step 16 a, Fourier space electromagnetic field equationsare set up in each of the hypothetical layers using the harmonicexpansion of the permittivity function. Second, in sub-step 16 b, theseFourier space equations are coupled using boundary conditions betweenharmonic expansion layers. Finally, in sub-step 16 c, the coupledFourier space equations are solved to provide the diffractedreflectivity. Each of these sub-steps is explained in detail below withreference to the corresponding step in the flow charts of FIGS. 6 and 7.

To set up the Fourier space electromagnetic field equations, it isconvenient to define the (2o+1)×(2o+1) Toeplitz-form, permittivityharmonics matrix E_(l) in step 14 c of FIG. 9. This permittivityharmonics matrix includes the harmonic components of the FourierExpansion of the permittivity ε_(l)(x) and is defined as:$\begin{matrix}{E_{l} = {\begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \cdots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \cdots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \cdots & ɛ_{l,{- {({{2o} - 2})}}} \\\cdots & \cdots & \cdots & \cdots & \cdots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \cdots & ɛ_{l,0}\end{bmatrix}.}} & \text{(1.1.7)}\end{matrix}$

A similar permittivity harmonics matrix is defined below in equation(2.1.4) which includes the harmonic components of the Fourier expansionof the inverse permittivity π_(l)(X).

As will be seen below, to perform a TE-polarization calculation whereoth-order harmonic components of the electric field {right arrow over(E)} and magnetic field {right arrow over (H)} are used, it is necessaryto use harmonics of the permittivity ε_(l,h) up to order 2o.

For the TE polarization, in the atmospheric layer the electric field{right arrow over (E)} is formulated (324) as $\begin{matrix}{{{\overset{\rightarrow}{E}}_{0,y} = {{\exp\left(  \right.} - {j\quad k_{0}{n_{0}\left( {{\sin \quad \theta \quad x} + {\cos \quad \theta \quad z}} \right)}} + {\sum\limits_{i}{R_{i}{\exp \left( {- {j\left( {{k_{xi}x} - {k_{0,{zi}}z}} \right)}} \right)}}}}},} & \text{(1.2.1)}\end{matrix}$

where the term on the left of the right-hand side of equation (1.2.1) isan incoming plane wave at an angle of incidence θ, the term on the rightof the right-hand side of equation (1.2.1) is a sum of reflected planewaves and R_(i) is the magnitude of the ith component of the reflectedwave, and the wave vectors k₀ and (k_(xi), k_(0,zi)) are given by$\begin{matrix}{{k_{0} = {\frac{2\pi}{\lambda} = {\omega \left( {\mu_{0}ɛ_{0}} \right)}^{1/2}}},} & \text{(1.2.2)} \\{{k_{xi} = {k_{0}\left( {{n_{0}{\sin (\theta)}} - {i\left( \frac{\lambda}{D} \right)}} \right)}},{and}} & \text{(1.2.3)} \\{k_{0,{zi}} = \left\{ \begin{matrix}{k_{0}\left( {n_{l}^{2} - \left( {k_{xi}/k_{0}} \right)^{2}} \right)}^{1/2} \\{{- j}\quad {k_{0}\left( {n_{l}^{2} - \left( {k_{xi}/k_{0}} \right)^{2}} \right)}^{1/2}}\end{matrix} \right.} & \text{(1.2.4)}\end{matrix}$

where the value of k_(0,zi) is chosen from equation (1.2.4), i.e., fromthe top or the bottom of the expression, to provideRe(k_(0,zi))−Im(k_(0,zi))>0. This insures that k_(0,zi) ² has a positivereal part, so that energy is conserved. It is easily confirmed that inthe atmospheric layer, the reflected wave vector (k_(xi), k_(0,zi)) hasa magnitude equal to that of the incoming wave vector k₀n₀. The magneticfield {right arrow over (H)} in the atmospheric layer is generated fromthe electric field {right arrow over (E)} by Maxwell's equation (1.3.1)provided below.

The x-components k_(xi) of the outgoing wave vectors satisfy the Floquetcondition (which is also called Bloch's Theorem, see Solid StatePhysics, N. W. Ashcrof and N. D. Mermin, Saunders College, Philadelphia,1976, pages 133-134) in each of the layers containing the periodicridges, and therefore, due to the boundary conditions, in theatmospheric layer and the substrate layer as well. That is, for a systemhaving an n-dimensional periodicity given by $\begin{matrix}{{{f\left( \overset{\rightarrow}{r} \right)} = {f\left( {\overset{\rightarrow}{r} + {\sum\limits_{i = 1}^{n}\quad {m_{i}{\overset{\rightarrow}{d}}_{i}}}} \right)}},} & \text{(1.2.5)}\end{matrix}$

where {right arrow over (d)}_(i) are the basis vectors of the periodicsystem, and m_(i) takes on positive and negative integer values, theFloquet condition requires that the wave vectors {right arrow over (k)}satisfy $\begin{matrix}{{\overset{\rightarrow}{k} = {{\overset{\rightarrow}{k}}_{0} + {2\pi {\sum\limits_{i = 1}^{n}\quad {m_{i}{\overset{\rightarrow}{b}}_{i}}}}}},} & \text{(1.2.6)}\end{matrix}$

where {right arrow over (b)}_(i) are the reciprocal lattice vectorsgiven by

({right arrow over (b)}_(i){right arrow over (d)}_(j))=δ_(ij),  (1.2.7)

{right arrow over (k)}₀ is the wave vector of a free-space solution, andδ_(ij) is the Kronecker delta function. In the case of the layers of theperiodic grating of FIGS. 6A and 6B which have the single reciprocallattice vector {right arrow over (b)} is {circumflex over (x)}/D,thereby providing the relationship of equation (1.2.3).

It may be noted that the formulation given above for the electric fieldin the atmospheric layer, although it is an expansion in terms of planewaves, is not determined via a Fourier transform of a real-spaceformulation. Rather, the formulation is produced (step 324) a prioribased on the Floquet condition and the requirements that both theincoming and outgoing radiation have wave vectors of magnitude n₀k₀.Similarly, the plane wave expansion for the electric field in thesubstrate layer is produced (step 324) a priori. In the substrate layer,the electric field {right arrow over (E)} is formulated (step 324) as atransmitted wave which is a sum of plane waves-where the x-componentsk_(xi) of the wave vectors (k_(xi), k_(0,zi)) satisfy the Floquetcondition, i.e., $\begin{matrix}{{{\overset{\rightarrow}{E}}_{L,y} = {\sum\limits_{i}{T_{i}{\exp \left( {- {j\left( {{k_{xi}x} + {k_{L,{zi}}\left( {z - {\sum\limits_{l = 1}^{L - 1}\quad t_{l}}} \right)}} \right)}} \right)}}}},} & \text{(1.2.8)}\end{matrix}$

where $\begin{matrix}{k_{L,{zi}} = \left\{ \begin{matrix}{k_{0}\left( {n_{L}^{2} - \left( {k_{xi}/k_{0}} \right)^{2}} \right)}^{1/2} \\{{- j}\quad {k_{0}\left( {n_{L}^{2} - \left( {k_{xi}/k_{0}} \right)^{2}} \right)}^{1/2}}\end{matrix} \right.} & \text{(1.2.9)}\end{matrix}$

where the value of k_(L,zi) is chosen from equation (1.2.9), i.e., fromthe top or the bottom of the expression, to provideRe(k_(L,zi))−Im(k_(L,zi))>0, insuring that energy is conserved.

The plane wave expansions for the electric and magnetic fields in theintermediate layers 225.1 through 225.(L−1) of FIG. 5 are also,referring again to FIG. 6, produced (step 334) a priori based on theFloquet condition. The electric field {right arrow over (E)}_(1,y) inthe lth layer is formulated (step 334) as a plane wave expansion alongthe direction of periodicity, {circumflex over (x)}, i.e.,$\begin{matrix}{{{\overset{\rightarrow}{E}}_{l,y} = {\sum\limits_{i}{{S_{l,{yi}}(z)}{\exp \left( {{- j}\quad k_{xi}x} \right)}}}},} & \text{(1.2.10)}\end{matrix}$

where S_(l,yi)(z) is the z-dependent electric field harmonic amplitudefor the lth layer and the ith harmonic. Similarly, the magnetic field{right arrow over (H)}_(l,y) in the lth layer is formulated (step 334)as a plane wave expansion along the direction of periodicity, {rightarrow over (x)}, i.e., $\begin{matrix}{{{\overset{\rightarrow}{H}}_{l,x} = {{- {j\left( \frac{ɛ_{0}}{\mu_{0}} \right)}^{1/2}}{\sum\limits_{i}{{U_{l,{xi}}(z)}{\exp \left( {{- j}\quad k_{xi}x} \right)}}}}},} & \text{(1.2.11)}\end{matrix}$

where U_(l,xi)(z) is the z-dependent magnetic field harmonic amplitudefor the lth layer and the ith harmonic.

According to Maxwell's equations, the electric and magnetic fieldswithin a layer are related by $\begin{matrix}{{{\overset{\rightarrow}{H}}_{l} = {\left( \frac{j}{{\omega\mu}_{0}} \right){\nabla{\times {\overset{\rightarrow}{E}}_{l}}}}},} & \text{(1.3.1)} \\{and} & \quad \\{{{\overset{\rightarrow}{E}}_{l}\left( \frac{- j}{{\omega ɛ}_{0}{ɛ_{l}(x)}} \right)}{\nabla{\times {{\overset{\rightarrow}{H}}_{l}.}}}} & \text{(1.3.2)}\end{matrix}$

As discussed above with respect to FIG. 9, in sub-step 16 b theseFourier space equations are coupled using boundary conditions betweenthe harmonic expansion layers. Applying (step 342) the first Maxwell'sequation (1.3.1) to equations (1.2.10) and (1.2.11) provides a firstrelationship between the electric and magnetic field harmonic amplitudesS_(l) and U_(l) of the lth layer: $\begin{matrix}{{\frac{\partial{S_{l,{yi}}(z)}}{\partial z} = {k_{0}U_{l,{xi}}}},} & \left( {1.3{.3}} \right)\end{matrix}$

Similarly, applying (step 341) the second Maxwell's equation (1.3.2) toequations (1.2.10) and (1.2.11), and taking advantage of therelationship $\begin{matrix}{{k_{xi} + \frac{2\pi \quad h}{D}} = k_{x{({i - h})}}} & \left( {1.3{.4}} \right)\end{matrix}$

which follows from equation (1.2.3), provides a second relationshipbetween the electric and magnetic field harmonic amplitudes S_(l) andU_(l) for the lth layer: $\begin{matrix}{\frac{\partial U_{l,{xi}}}{\partial z} = {{\left( \frac{k_{xi}^{2}}{k_{0}} \right)S_{l,{yi}}} - {k_{0}{\sum\limits_{p}\quad {ɛ_{({i - p})}{S_{l,{yp}}.}}}}}} & \left( {1.3{.5}} \right)\end{matrix}$

While equation (1.3.3) only couples harmonic amplitudes of the sameorder i, equation (1.3.5) couples harmonic amplitudes S_(l) and U_(l)between harmonic orders. In equation (1.3.5), permittivity harmonicsε_(i) from order −2o to +2o are required to couple harmonic amplitudesS_(l) and U_(l) of orders between −o and +o.

Combining equations (1.3.3) and (1.3.5) and truncating the calculationto order o in the harmonic amplitude S provides (step 345) asecond-order differential matrix equation having the form of a waveequation, i.e., $\begin{matrix}{{\left\lbrack \frac{\partial^{2}S_{l,y}}{\partial z^{\prime 2}} \right\rbrack = {\left\lbrack A_{l} \right\rbrack \left\lbrack S_{l.y} \right\rbrack}},} & \left( {1.3{.6}} \right)\end{matrix}$

z′ =k₀ z, the wave-vector matrix [A_(l)] is defined as

[A₁]=[K_(x)]²−[E_(l] ,)  (1.3.7)

where [K_(x)] is a diagonal matrix with the (i,i) element being equal to(k_(xi)/k₀), the permittivity harmonics matrix [E_(l)] is defined abovein equation (1.1.4), and [S_(l,y)] and [∂²S_(l,y)/∂z′²] are columnvectors with indices i running from −o to +o, i.e., $\begin{matrix}{{\left\lbrack S_{l,y} \right\rbrack = \begin{bmatrix}S_{l,y,{({- 0})}} \\\vdots \\S_{l,y,0} \\\vdots \\S_{l,y,0}\end{bmatrix}},} & \text{(1.3.8)}\end{matrix}$

By writing (step 350) the homogeneous solution of equation (1.3.6) as anexpansion in pairs of exponentials, i.e., $\begin{matrix}{{{S_{l,{yi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}\quad {w_{l,i,m}\left\lbrack {{{c1}_{l,m}{\exp \left( {{- k_{0}}q_{l,m}z} \right)}} + {{c2}_{l,m}{\exp \left( {k_{0}{q_{l,m}\left( {z - t_{l}} \right)}} \right)}}} \right\rbrack}}},} & \text{(1.3.9)}\end{matrix}$

its functional form is maintained upon second-order differentiation byz′, thereby taking the form of an eigen equation. Solution (step 347) ofthe eigen equation

[A_(l)|W_(l)]=[τ_(l)][W_(l)] ,  (1.3.10)

provides (step 348) a diagonal eigenvalue matrix [τ_(l)] formed from theeigenvalues τ_(l,m) of the wave-vector matrix [A_(l)], and aneigenvector matrix [W_(l)] of entries w_(l,i,m), where w_(l,i,m) is theith entry of the mth eigenvector of [A_(l)]. A diagonal root-eigenvaluematrix [Q_(l)] is defined to be diagonal entries q_(l,i) which are thepositive real portion of the square roots of the eigenvalues τ_(l,i).The constants c1 and c2 are, as yet, undetermined.

By applying equation (1.3.3) to equation (1.3.9) it is found that$\begin{matrix}{{U_{l,{xi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}\quad {v_{l,i,m}\left\lbrack {{{- {c1}_{l,m}}{\exp \left( {{- k_{0}}q_{l,m}z} \right)}} + {{c2}_{l,m}{\exp \left( {k_{0}{q_{l,m}\left( {z - t_{l}} \right)}} \right)}}} \right\rbrack}}} & \text{(1.3.11)}\end{matrix}$

where v_(l,i,m)=q_(l,m)w_(l,i,m). The matrix [V_(l)], to be used below,is composed of entries v_(l,i,m).

The constants c1 and c2 in the homogeneous solutions of equations(1.3.9) and (1.3.11) are determined by applying (step 355) therequirement that the tangential electric and magnetic fields becontinuous at the boundary between each pair of adjacent layers 225.land 225.(l+1). At the boundary between the atmospheric layer and thefirst layer 225.1, continuity of the electric field E_(y) and themagnetic field H_(x) requires $\begin{matrix}{{\begin{bmatrix}\delta_{i0} \\{{jn}_{0}{\cos (\theta)}\delta_{i0}}\end{bmatrix} + {\begin{bmatrix}I \\{- {jY}_{0}}\end{bmatrix}R}} = {\begin{bmatrix}W_{1} & {W_{1}X_{1}} \\V_{1} & {{- V_{1}}X_{1}}\end{bmatrix}\quad\begin{bmatrix}{c1}_{1} \\{c2}_{1}\end{bmatrix}}} & \left( {1.4{.1}} \right)\end{matrix}$

where Y₀ is a diagonal matrix with entries (k_(0,zi)/ k₀), X_(l) is adiagonal layer-translation matrix with elements exp(−k₀ q_(l,m) t_(l)),R is a vector consisting of entries from R_(−o) to R_(+o) and c1 ₁ andc2 ₁ are vectors consisting of entries from c1 _(1.0) and c1 _(1,2o+1),and c2 _(1,0) and c2 _(1,2o+1), respectively. The top half of matrixequation (1.4.1) provides matching of the electric field E_(y) acrossthe boundary of the atmospheric layer 225.0 and the first layer 225.1,the bottom half of matrix equation (1.4.1) provides matching of themagnetic field H_(x) across the layer boundary between layer 225.0 andlayer 125.1, the vector on the far left is the contribution from theincoming radiation, shown in FIG. 4, in the atmospheric layer 701 ofFIG. 5, the second vector on the left is the contribution from thereflected radiation 132, shown in FIG. 4, in the atmospheric layer 701of FIG. 5, and the portion on the right represents the fields E_(y) andH_(x) in the first layer 225.1 of FIG. 5.

At the boundary between adjacent intermediate layers 225.l and225.(l+1), continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{l - 1}X_{l - 1}} & W_{l - 1} \\{W_{l - 1}X_{l - 1}} & {- V_{l - 1}}\end{bmatrix}\quad\begin{bmatrix}{c1}_{l - 1} \\{c2}_{l - 1}\end{bmatrix}} = {\begin{bmatrix}W_{l} & {W_{l}X_{l}} \\V_{l} & {{- V_{l}}X_{l}}\end{bmatrix}\quad\begin{bmatrix}{c1}_{l} \\{c2}_{l}\end{bmatrix}}},} & \left( {1.4{.2}} \right)\end{matrix}$

where the top and bottom halves of the vector equation provide matchingof the electric field E_(y) and the magnetic field H_(x), respectively,across the l−1/l layer boundary.

At the boundary between the (L−l)th layer 225.(L−1) and the substratelayer, continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{L - 1}X_{L - 1}} & W_{L - 1} \\{V_{L - 1}X_{L - 1}} & {- V_{L - 1}}\end{bmatrix}\quad\begin{bmatrix}{c1}_{L - 1} \\{c2}_{L - 1}\end{bmatrix}} = {\begin{bmatrix}I \\{jY}_{L}\end{bmatrix}T}},} & \left( {1.4{.3}} \right)\end{matrix}$

where, as above, the top and bottom halves of the vector equationprovides matching of the electric field E_(y) and the magnetic fieldH_(x), respectively. In contrast with equation (1.4.1), there is only asingle term on the right since there is no incident radiation in thesubstrate.

Referring again to FIG. 6, matrix equation (1.4.1), matrix equation(1.4.3), and the (L−1) matrix equations (1.4.2) can be combined (step360) to provide a boundary-matched system matrix equation$\begin{matrix}{{{\begin{bmatrix}\quad & {- I} & W_{1} & {W_{1}X_{1}} & 0 & {0\quad \ldots} & \quad & \quad \\\quad & {j\quad Y_{0}} & V_{1} & {- {VX}} & 0 & {0\quad \ldots} & \quad & \quad \\0 & \quad & {{- W_{1}}X_{1}} & {- W_{1}} & W_{2} & {W_{2}X_{2}} & 0 & {0\quad \ldots} \\0 & \quad & {{- V_{1}}X_{1}} & V_{1} & V_{2} & {{- V_{2}}X_{2}} & 0 & {0\quad \ldots} \\\quad & 0 & 0 & ⋰ & \quad & ⋰ & \quad & \vdots \\\quad & 0 & 0 & \quad & \quad & \quad & \quad & \quad \\\quad & \quad & \quad & \cdots & \quad & {{- W_{L - 1}}X_{L - 1}} & {- W_{L - 1}} & I \\\quad & \quad & \quad & \quad & \quad & {{- V_{L - 1}}X_{L - 1}} & V_{L - 1} & {j\quad Y_{L}}\end{bmatrix}\begin{bmatrix}R \\{c1}_{1} \\{c2}_{1} \\\vdots \\\vdots \\{c1}_{L - 1} \\{c2}_{L - 1} \\T\end{bmatrix}} = \begin{bmatrix}\delta_{i0} \\{{j\delta}_{i0}n_{0}{\cos (\theta)}} \\0 \\\vdots \\\quad \\\quad \\\vdots \\0\end{bmatrix}},} & \text{(1.4.4)}\end{matrix}$

As is well understood by those skilled in the art, this boundary-matchedsystem matrix equation (1.4.4) may be solved (step 365) (sub-step 16 cin the flow chart of FIG. 9) to provide the reflectivity R_(i) for eachharmonic order i. (Alternatively, the partial-solution approachdescribed in “Stable Implementation of the Rigorous Coupled-WaveAnalysis for Surface-Relief Dielectric Gratings: Enhanced TransmittanceMatrix Approach”, E. B. Grann and D. A. Pommet, J Opt. Soc. Am. A, vol.12, 1077-1086, May 1995, can be applied to calculate either thediffracted reflectivity R or the diffracted transmittance T.)

As noted above any planar polarization is a combination of in-phase TEand TM polarizations. The method of the present invention can be appliedto any polarization which is a superposition of TE and TM polarizationsby computing the diffraction of the TE and TM components separately andsumming them.

The method 400 of calculation for the diffracted reflectivity ofTM-polarized incident electromagnetic radiation shown in FIG. 7parallels that method 300 described above and shown in FIG. 6 for thediffracted reflectivity of TE-polarized incident electromagneticradiation. Referring to FIG. 4, for TM-polarized incident radiation 631the electric field vector {right arrow over (E)} is in the plane ofincidence 640, and the magnetic field vector {right arrow over (H)} isperpendicular to the plane of incidence 640. (The similarity in theTE-and TM-polarization RCWA calculations and the application of thepresent invention motivates the use of the term ‘electromagnetic field’in the present specification to refer generically to either or both theelectric field and/or the magnetic field of the electromagneticradiation.)

As above, once the permittivity ε_(l)(x) is determined or acquired (step410), the permittivity harmonics ε_(l,i) are determined (step 412) usingFourier transforms according to equations (1.1.2) and (1.1.3), and thepermittivity harmonics matrix E_(l) is assembled as per equation(1.1.4). In the case of TM-polarized incident radiation, it has beenfound that the accuracy of the calculation may be improved byformulating the calculations using inverse-permittivity harmonicsπ_(l,i), since this will involve the inversion of matrices which areless singular. In particular, the one-dimensional Fourier expansion(step 412) for the inverse of the permittivity ε_(l)(x) of the lth layeris given by $\begin{matrix}{\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}{\exp \left( {j\frac{2{\pi i}}{D}x} \right)}}}} & \left( {2.1{.1}} \right)\end{matrix}$

Therefore, via the inverse Fourier transform this provides$\begin{matrix}{{\pi_{l,0} = {{\sum\limits_{k = 1}^{r}\quad {\frac{1}{n_{k}^{2}}x_{k}}} - \frac{x_{k - 1}}{D}}},} & \left( {2.1{.2}} \right)\end{matrix}$

and for i not equal to zero, $\begin{matrix}{\pi_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{{- j}\quad {i2\pi}}\frac{1}{n_{k}^{2}}\left( {\left( {{\cos \left( {\frac{2\quad \pi \quad i}{D}x_{k}} \right)} - {\cos \left( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} - {\sin \left( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} \right)}} \right)}} \right)}}} & \left( {2.1{.3}} \right)\end{matrix}$

where the sum is over the number r of borders and n_(k) is the index ofrefraction of the material between the kth and the (k−1)th border and jis the imaginary number defined as the square root of −1.

As noted above with respect to equations (1.1.1) through (1.1.3),(1.1.2.1) and (1.1.3.1), by describing a periodic grating such as asemiconductor device by dividing the grating into layers as discussedabove and shown in FIG. 5, further subdividing the layers into areaswhere the layers intersect with one of the materials, and expanding theinverse permittivity of a periodic grating such as grating 700 as shownin equations (2.1.1) through (2.1.3), as described below, a method andsystem in accordance with the present invention can be used to determinethe diffracted reflectivity of a semiconductor device having transverseor other defects, such as a polymer coating residue formed from anetching operation and illustrated in FIG. 2B, resulting in three or morematerials per layer in a periodic direction. Additionally, the methoddisclosed herein of describing a periodic grating such as asemiconductor device by dividing the grating into layers as discussedabove and shown in FIG. 5, and expanding the inverse permittivity of aperiodic grating such as grating 700 as shown in equations (2.1.1)through (2.1.3) can be used with any optical profilometry formalism fordetermining a diffracted reflectivity that uses a Fourier transform ofthe inverse permittivity.

The inverse-permittivity harmonics matrix P_(l) is defined as$\begin{matrix}{{P_{1} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \cdots & \pi_{l,{2o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \cdots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \cdots & \pi_{l,{- {({{2o} - 2})}}} \\\cdots & \cdots & \cdots & \cdots & \cdots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \cdots & \pi_{l,0}\end{bmatrix}},} & \text{(2.1.4)}\end{matrix}$

where 2o is the maximum harmonic order of the inverse permittivityπ_(l,i) used in the calculation. As with the case of the TEpolarization, for electromagnetic fields {right arrow over (E)} and{right arrow over (H)} calculated to order o it is necessary to useharmonic components of the permittivity ε_(l,i) and inverse permittivityπ_(l,i) to order 2o.

In the atmospheric layer the magnetic field {right arrow over (H)} isformulated (step 424) a priori as a plane wave incoming at an angle ofincidence θ, and a reflected wave which is a sum of plane waves havingwave vectors (k_(xi), k_(0,zi)) satisfying the Floquet condition,equation (1.2.6). In particular, $\begin{matrix}{{{\overset{\rightarrow}{H}}_{0,y} = {{\exp\left(  \right.} - {j\quad k_{0}{n_{0}\left( {{\sin \quad \theta \quad x} + {\cos \quad \theta \quad z}} \right)}} + {\sum\limits_{i}{R_{i}{\exp \left( {- {j\left( {{k_{xi}x} - {k_{0,{zi}}z}} \right)}} \right)}}}}},} & \text{(2.2.1)}\end{matrix}$

where the term on the left of the right-hand side of the equation is theincoming plane wave, and R_(i) is the magnitude of the ith component ofthe reflected wave. The wave vectors k₀ and (k_(xi), k_(0,zi)) are givenby equations (1.2.2), (1.2.3), and (1.2.4) above, and, referring now toFIG. 5, the magnetic field {right arrow over (H)} in the atmosphericlayer 701 is generated from the electric field {right arrow over (E)} byMaxwell's equation (1.3.2). In the substrate layer 710 the magneticfield {right arrow over (H)} is, as shown in FIG. 6, formulated (step424) as a transmitted wave which is a sum of plane waves where the wavevectors (k_(xi), k_(0,zi)) satisfy the Floquet condition, equation(1.2.6), i.e., $\begin{matrix}{{{\overset{\rightarrow}{H}}_{L,y} = {\sum\limits_{i}{T_{i}{\exp \left( {- {j\left( {{k_{xi}x} + {k_{L,{zi}}\left( {z - {\sum\limits_{l = 1}^{L - 1}\quad t_{l}}} \right)}} \right)}} \right)}}}},} & \text{(2.2.2)}\end{matrix}$

where k_(L,zi) is defined in equation (1.2.9). Again based on theFloquet condition, the magnetic field {right arrow over (H)}_(l,y) inthe lth layer is formulated (step 434) as a plane wave expansion alongthe direction of periodicity, {circumflex over (x)}, i.e.,$\begin{matrix}{{{\overset{\rightarrow}{H}}_{l,y} = {\sum\limits_{i}{{U_{l,{yi}}(z)}{\exp \left( {{- j}\quad k_{xi}x} \right)}}}},} & \text{(2.2.3)}\end{matrix}$

where U_(l,yi)(z) is the z-dependent magnetic field harmonic amplitudefor the lth layer and the ith harmonic. Similarly, the electric field{right arrow over (E)}_(l,x) in the lth layer is formulated (step 434)as a plane wave expansion along the direction of periodicity, i.e.,$\begin{matrix}{{{\overset{\rightarrow}{E}}_{l,x} = {{j\left( \frac{\mu_{0}}{ɛ_{0}} \right)}^{1/2}{\sum\limits_{i}{{S_{l,{xi}}(z)}{\exp \left( {{- j}\quad k_{xi}x} \right)}}}}},} & \text{(2.2.4)}\end{matrix}$

where S_(l,xi)(z) is the z-dependent electric field harmonic amplitudefor the lth layer and the ith harmonic.

Substituting equations (2.2.3) and (2.2.4) into Maxwell's equation(1.3.2) provides(step 441) a first relationship between the electric andmagnetic field harmonic amplitudes S_(l) and U_(l) for the lth layer:$\begin{matrix}{\frac{\partial\left\lbrack U_{l,{yi}} \right\rbrack}{\partial z^{\prime}} = {{\left\lbrack E_{l} \right\rbrack \left\lbrack S_{l,{xi}} \right\rbrack}.}} & \text{(2.3.1)}\end{matrix}$

Similarly, substituting (2.2.3) and (2.2.4) into Maxwell's equation(1.3.1) provides (step 442) a second relationship between the electricand magnetic field harmonic amplitudes S_(l) and U_(l) for the lthlayer: $\begin{matrix}{\frac{\partial\left\lbrack S_{l,{xi}} \right\rbrack}{\partial z^{\prime}} = {{\left( {{{\left\lbrack K_{x} \right\rbrack \left\lbrack P_{l} \right\rbrack}\left\lbrack K_{x} \right\rbrack} - \lbrack I\rbrack} \right)\left\lbrack U_{l,y} \right\rbrack}.}} & \text{(2.3.2)}\end{matrix}$

where, as above, K_(x) is a diagonal matrix with the (i,i) element beingequal to (k_(xi)/k₀). In contrast with equations (1.3.3) and (1.3.5)from the TE-polarization calculation, non-diagonal matrices in bothequation (2.3.1) and equation (2.3.2) couple harmonic amplitudes S_(l)and U_(l) between harmonic orders.

Combining equations (2.3.1) and (2.3.2) provides a second-orderdifferential wave equation $\begin{matrix}{{\left\lbrack \frac{\partial^{2}U_{l,y}}{\partial z^{\prime 2}} \right\rbrack = {\left\{ {\left\lbrack E_{l} \right\rbrack \left( {{{\left\lbrack K_{x} \right\rbrack \left\lbrack P_{l} \right\rbrack}\left\lbrack K_{x} \right\rbrack} - \lbrack I\rbrack} \right)} \right\} \left\lbrack U_{l,y} \right\rbrack}},} & \text{(2.3.3)}\end{matrix}$

where [U_(l,y)] and [∂²U_(l,y)/∂z′²] are column vectors with indicesrunning from −o to +o, and the permittivity harmonics [E_(l)] is definedabove in equation (1.1.7), and z′=k₀z. The wave-vector matrix [A_(l)]for equation (2.3.3) is defined as

[A _(l) ]=[E _(l)]([K _(x) ][P _(l) ][K _(x) ]−[I]).  (2.3.4)

If an infinite number of harmonics could be used, then the inverse ofthe permittivity harmonics matrix [E_(l)] would be equal to theinverse-permittivity harmonics matrix [P_(l)], and vice versa, i.e.,[E_(l)]⁻¹=[P_(l)], and [P_(l)]⁻¹=[E_(l)]. However, the equality does nothold when a finite number o of harmonics is used, and for finite o thesingularity of the matrices [E_(l)]⁻¹ and [P_(l)], and the singularityof the matrices [P_(l)]⁻¹ and [E_(l)], will generally differ. In fact,it has been found that the accuracy of RCWA calculations will varydepending on whether the wave-vector matrix [A_(l)] is defined as inequation (2.3.4), or

[A _(l) ]=[P _(l)]⁻¹([K _(x) ][E _(l)]⁻¹ [K _(x) ]−[I]),  (2.3.5)

or

[A _(l) ]=[E _(l)]([K _(x) ][E _(l)]⁻¹ [K _(x) ]−[I]).  (2.3.6)

It should also be understood that although the case where

[A _(l) ]=[P _(l)]⁻¹([K _(x) ][P _(l) ][K _(x) ]−[I])  (2.3.6′)

does not typically provide convergence which is as good as theformulations of equation (2.3.5) and (2.3.6), the present invention mayalso be applied to the formulation of equation (2.3.6′).

Regardless of which of the three formulations, equations (2.3.4),(2.3.5) or (2.3.6), for the wave-vector matrix [A_(l)] is used, thesolution of equation (2.3.3) is performed by writing (step 450) thehomogeneous solution for the magnetic field harmonic amplitude U_(l) asan expansion in pairs of exponentials, i.e., $\begin{matrix}{{U_{l,{yi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}\quad {{w_{l,i,m}\left\lbrack {{{c1}_{l,m}{\exp \left( {{- k_{o}}q_{l,m}z} \right)}} + {{c2}_{l,m}{\exp \left( {k_{o}{q_{l,m}\left( {z - t_{l}} \right)}} \right)}}} \right\rbrack}.}}} & \text{(2.3.7)}\end{matrix}$

since its functional from is maintained upon second-orderdifferentiation by z′, and equation (2.3.3) becomes an eigen equation.Solution (step 447) of the eigen equation

[A _(l) ][W _(l)]=[τ_(l) ][W _(l)],  (2.3.8)

provides (step 448) an eigenvector matrix [W_(l)] formed from theeigenvectors w_(l,i) of the wave-vector matrix [A_(l)], and a diagonaleigenvalue matrix [τ_(l)] formed from the eigenvalues τ_(l,i) of thewave-vector matrix [A_(l)]. A diagonal root-eigenvalue matrix [Q_(l)] isformed of diagonal entries q_(l.i) which are the positive real portionof the square roots of the eigenvalues τ_(l,i). The constants c1 and c2of equation (2.3.7) are, as yet, undetermined.

By applying equation (1.3.3) to equation (2.3.5) it is found that$\begin{matrix}{{S_{l,{xi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}\quad {v_{l,i,m}\left\lbrack {{{- {c1}_{l,m}}{\exp \left( {{- k_{0}}q_{l,m}z} \right)}} + {{c2}_{l,m}{\exp \left( {k_{0}{q_{l,m}\left( {z - t_{l}} \right)}} \right)}}} \right\rbrack}}} & \text{(2.3.9)}\end{matrix}$

where the vectors v_(l,i) form a matrix [V_(l)] defined as$\begin{matrix}{{\lbrack V\rbrack = {{\lbrack E\rbrack^{- 1}\lbrack W\rbrack}\lbrack Q\rbrack}}\quad {{{{when}\quad\lbrack A\rbrack}\quad {is}\quad {defined}\quad {as}\quad {in}\quad {equation}\quad \left( {2.3{.4}} \right)},}} & \left( {2.3{.10}} \right) \\{{\lbrack V\rbrack = {{\lbrack P\rbrack \lbrack W\rbrack}\lbrack Q\rbrack}}\quad {{{{when}\quad\lbrack A\rbrack}\quad {is}\quad {defined}\quad {as}\quad {in}\quad {equation}\quad \left( {2.3{.5}} \right)},}} & \left( {2.3{.11}} \right) \\{{\lbrack V\rbrack = {{\lbrack E\rbrack^{- 1}\lbrack W\rbrack}\lbrack Q\rbrack}}\quad \quad {{{when}\quad\lbrack A\rbrack}\quad {is}\quad {defined}\quad {as}\quad {in}\quad {equation}\quad {\left( {2.3{.6}} \right).}}} & \left( {2.3{.12}} \right)\end{matrix}$

The formulation of equations (2.3.5) and (2.3.11) typically has improvedconvergence performance (see P. Lalanne and G. M. Morris, “HighlyImproved Convergence of the Coupled-Wave Method for TM Polarization”, J.Opt. Soc. Am. A, 779-784, 1996; and L. Li and C. Haggans, “Convergenceof the coupled-wave method for metallic lamellar diffraction gratings”,J. Opt. Soc. Am. A, 1184-1189, June 1993) relative to the formulation ofequations (2.3.4) and (2.3.11) (see M. G. Moharam and T. K. Gaylord,“Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction”, J. Opt.Soc. Am., vol. 71, 811-818, July 1981).

The constants c1 and c2 in the homogeneous solutions of equations(2.3.7) and (2.3.9) are determined by applying (step 455) therequirement that the tangential electric andtangential magnetic fieldsbe continuous at the boundary between each pair of adjacent layers(125.l)/(125.(l+1)), when the materials in each layer non-conductive.The calculation of the present specification is straightforwardlymodified to circumstances involving conductive materials, and theapplication of the method of the present invention to periodic gratingswhich include conductive materials is considered to be within the scopeof the present invention. Referring to FIG. 5, at the boundary betweenthe atmospheric layer 701 and the first layer 225.1, continuity of themagnetic field H_(y) and the electric field E_(x) requires$\begin{matrix}{{\begin{bmatrix}\delta_{i0} \\{{{jcos}(\theta)}{\delta_{i0}/n_{0}}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\quad Z_{0}}\end{bmatrix}R}} = {\begin{bmatrix}W_{1} & {W_{1}X_{1}} \\V_{1} & {{- V_{1}}X_{1}}\end{bmatrix}\begin{bmatrix}{c1}_{1} \\{c2}_{1}\end{bmatrix}}} & \text{(2.4.1)}\end{matrix}$

where Z₀ is a diagonal matrix with entries (k_(0,zi)/n₀ ² k₀), X_(l) isa diagonal matrix with elements exp(−k₀ q_(l,m) t_(l)), the top half ofthe vector equation provides matching of the magnetic field H_(y) acrossthe layer boundary, the bottom half of the vector equation providesmatching of the electric field E_(x) across the layer boundary, thevector on the far left is the contribution from incoming radiation inthe atmospheric layer 701, the second vector on the left is thecontribution from reflected radiation in the atmospheric layer 701, andthe portion on the right represents the fields H_(y) and E_(x) in thefirst layer 225.1.

At the boundary between adjacent intermediate layers 225.1 and225.(l+1), continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{l - 1}X_{l - 1}} & W_{l - 1} \\{W_{l - 1}X_{l - 1}} & {- V_{l - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{l - 1} \\{c2}_{l - 1}\end{bmatrix}} = {\begin{bmatrix}W_{l} & {W_{l}X_{l}} \\V_{l} & {{- V_{l}}W_{l}}\end{bmatrix}\begin{bmatrix}{c1}_{l} \\{c2}_{l}\end{bmatrix}}},} & \text{(2.4.2)}\end{matrix}$

where the top and bottom halves of the vector equation provides matchingof the magnetic field H_(y) and the electric field E_(x), respectively,across the layer boundary.

At the boundary between the (L−1)th layer 225.(L−1) and the substratelayer 710, continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{L - 1}X_{L - 1}} & W_{L - 1} \\{V_{L - 1}X_{L - 1}} & {- V_{L - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{L - 1} \\{c2}_{L - 1}\end{bmatrix}} = {\begin{bmatrix}I \\{j\quad Z_{L}}\end{bmatrix}T}},} & \text{(2.4.3)}\end{matrix}$

where, as above, the top and bottom halves of the vector equationprovides matching of the magnetic field H_(y) and the electric fieldE_(x), respectively. In contrast with equation (2.4.1), there is only asingle term on the right in equation (2.4.3) since there is no incidentradiation in the substrate 710.

Matrix equation (2.4.1), matrix equation (2.4.3), and the (L−1) matrixequations (2.4.2) can be combined (step 460) to provide aboundary-matched system matrix equation $\begin{matrix}{{{\begin{bmatrix}\quad & {- I} & W_{1} & {W_{1}X_{1}} & 0 & {0\quad \ldots} & \quad & \quad \\\quad & {j\quad Z_{0}} & V_{1} & {- {VX}} & 0 & {0\quad \ldots} & \quad & \quad \\0 & \quad & {{- W_{1}}X_{1}} & {- W_{1}} & W_{2} & {W_{2}X_{2}} & 0 & {0\quad \ldots} \\0 & \quad & {{- V_{1}}X_{1}} & V_{1} & V_{2} & {{- V_{2}}X_{2}} & 0 & {0\quad \ldots} \\\quad & 0 & 0 & ⋰ & \quad & ⋰ & \quad & \vdots \\\quad & 0 & 0 & \quad & \quad & \quad & \quad & \quad \\\quad & \quad & \quad & \cdots & \quad & {{- W_{L - 1}}X_{L - 1}} & {- W_{L - 1}} & I \\\quad & \quad & \quad & \quad & \quad & {{- V_{L - 1}}X_{L - 1}} & V_{L - 1} & {j\quad Z_{L}}\end{bmatrix}\begin{bmatrix}R \\{c1}_{1} \\{c2}_{1} \\\vdots \\\vdots \\{c1}_{L - 1} \\{c2}_{L - 1} \\T\end{bmatrix}} = \begin{bmatrix}\delta_{i0} \\{{j\delta}_{i0}{{\cos (\theta)}/n_{0}}} \\0 \\\vdots \\\quad \\\quad \\\vdots \\0\end{bmatrix}},} & \text{(2.4.4)}\end{matrix}$

As is well understood by those skilled in the art, the boundary-matchedsystem matrix equation (2.4.4) may be solved (step 465) to provide thereflectivity R for each harmonic order i. (Alternatively, thepartial-solution approach described in “Stable Implementation of theRigorous Coupled-Wave Analysis for Surface-Relief Dielectric Gratings:Enhanced Transmittance Matrix Approach”, E. B. Grann and D. A. Pommet, JOpt. Soc. Am. A, vol. 12, 1077-1086, May 1995, can be applied tocalculate either the diffracted reflectivity R or the diffractedtransmittance T.)

The matrix on the left in boundary-matched system matrix equations(1.4.4) and (2.4.4) is a square non-Hermetian complex matrix which issparse (i.e., most of its entries are zero), and is of constant blockconstruction (i.e., it is an array of sub-matrices of uniform size). Thematrix can be stored in a database to provide computer access forsolving for the diffracted reflectivity using numerical methods. As iswell known by those skilled in the art, the matrix can be stored usingthe constant block compressed sparse row data structure (BSR) method(see S. Carney, M. Heroux, G. Li, R. Pozo, K. Remington and K. Wu, “ARevised Proposal for a Sparse BLAS Toolkit,”http://www.netlib.org,1996). In particular, for a matrix composed of a square array of squaresub-matrices, the BSR method uses five descriptors:

B₁₃LDA is the dimension of the array of sub-matrices;

O is the dimension of the sub-matrices;

VAL is a vector of the non-zero sub-matrices starting from the leftmostnon-zero matrix in the top row (assuming that there is a non-zero matrixin the top row), and continuing on from left to right, and top tobottom, to the rightmost non-zero matrix in the bottom row (assumingthat there is a non-zero matrix in the bottom row).

COL₁₃IND is a vector of the column indices of the sub-matrices in theVAL vector; and

ROW₁₃PTR is a vector of pointers to those sub-matrices in VAL which arethe first non-zero sub-matrices in each row.

For example, for the left-hand matrix of equation (1.4.4), B₁₃ LDA has avalue of 2L, O has a value of 2o+1, the entries of VAL are (−I, W_(l),W_(l)X_(l), jY₀, V_(l), −V_(l)X_(l), −W_(l)X1l, −W_(l), W₂, W₂X₂,−V_(l)X_(l), V_(l), V₂, . . . ), the entries of COL₁₃IND are (1, 2, 3,1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, . . . ), and the entries of ROW₁₃PTRare (1, 4, 7, 11, . . . ).

As is well-known in the art of the solution of matrix equations, thesquareness and sparseness of the left-hand matrices of equations (1.4.4)and (2.4.4) are used to advantage by solving equations (1.4.4) and(2.4.4) using the Blocked Gaussian Elimination (BGE) algorithm. The BGEalgorithm is derived from the standard Gaussian Elimination algorithm(see, for example, Numerical Recipes, W. H. Press, B. P. Flannery, S. A.Teukolsky, and W. T. Vetterling, Cambridge University Press, Cambridge,1986, pp. 29-38) by the substitution of sub-matrices for scalars.According to the Gaussian Elimination method, the left-hand matrix ofequation (1.4.4) or (2.4.4) is decomposed into the product of a lowertriangular matrix [L], and an upper triangular matrix [U], to provide anequation of the form

[L][U][x]=[b],  (3.1.1)

and then the two triangular systems [U][x]=[y] and [L] [y]=[b] aresolved to obtain the solution [x]=[U]⁻¹ [L]⁻¹ [b], where, as perequations (1.4.4) and (2.4.4), [x] includes the diffracted reflectivityR.

It should be noted that although the invention has been described interm of a method, as per FIGS. 6, 7 and 9, the invention mayalternatively be viewed as an apparatus or system. Specifically, a shownin FIG. 10, the method of the present invention is preferablyimplemented on a computer system 900. Computer system 900 preferablyincludes information input/output (I/O) equipment 905, which isinterfaced to a computer 910. Computer 910 includes a central processingunit (CPU) 915, a cache memory 920 and a persistent memory 925, whichpreferably includes a hard disk, floppy disk or other computer readablemedium. The I/O equipment 905 typically includes a keyboard 902 and amouse 904 for the input of information, a display device 901 and aprinter 903. Many variations on computer system 900 are to be consideredwithin the scope of the present invention, including, withoutlimitation, systems with multiple I/O devices, multiple processors witha single computer, multiple computers connected by Internet linkages,and multiple computers connected by a local area network.

As is well understood by those skilled in the art, software computercode for implementing the steps of the method of the present inventionillustrated in FIGS. 6, 7 and 9 and discussed in detail above can bestored in persistent memory 925. CPU 915 can then execute the steps ofthe method of the present invention and store results of executing thesteps in cache memory 920 for completing diffracted reflectivitycalculations as discussed above.

Referring to FIG. 4, it should also be understood that the presentinvention is applicable to off-axis or conical incident radiation 631(i.e., the case where φ≈0 and the plane of incidence 640 is not alignedwith the direction of periodicity, {circumflex over (x)}, of thegrating). The above exposition is straight forwardly adapted to theoff-axis case since, as can be seen in “Rigorous Coupled-Wave Analysisof Planar-Grating Diffraction,” M.

G. Moharam and T. K. Gaylord, J. Opt. Soc. Am., vol. 71, 811-818, July1981, the differential equations for the electromagnetic fields in eachlayer have homogeneous solutions with coefficients and factors that areonly dependent on intra-layer parameters and incident-radiationparameters.

It is also important to understand that, although the present inventionhas been described in terms of its application to the rigorouscoupled-wave method of calculating the diffraction of radiation, themethod of the present invention may be applied to any opticalprofilometry formalism where the system is divided into layers. Theforegoing descriptions of specific embodiments of the present inventionhave been presented for purposes of illustration and description. Theyare not intended to be exhaustive or to limit the invention to theprecise forms disclosed, and it should be understood that manymodifications and variations are possible in light of the aboveteaching. The embodiments were chosen and described in order to bestexplain the principles of the invention and its practical application,to thereby enable others skilled in the art to best utilize theinvention and various embodiments with various modifications as aresuited to the particular use contemplated. Many other variations arealso to be considered within the scope of the present invention.

Additionally, the calculation of the present specification is applicableto circumstances involving conductive materials, or non-conductivematerials, or both, and the application of the method of the presentinvention to periodic gratings which include conductive materials isconsidered to be within the scope of the present invention; theeigenvectors and eigenvalues of the matrix [A] may be calculated usinganother technique; the layer boundaries need not be planar andexpansions other than Fourier expansions, such as Bessel or Legendreexpansions, may be applied; the “ridges” and “troughs” of the periodicgrating may be ill-defined; the method of the present invention may beapplied to gratings having two-dimensional periodicity; the method ofthe present invention may be applied to any polarization which is asuperposition of TE and TM polarizations; the ridged structure of theperiodic grating may be mounted on one or more layers of films depositedon the substrate; the method of the present invention may be used fordiffractive analysis of lithographic masks or reticles; the method ofthe present invention may be applied to sound incident on a periodicgrating; the method of the present invention may be applied to medicalimaging techniques using incident sound or electromagnetic waves; themethod of the present invention may be applied to assist in real-timetracking of fabrication processes; the gratings may be made by ruling,blazing or etching; the method of the present invention may be utilizedin the field of optical analog computing, volume holographic gratings,holographic neural networks, holographic data storage, holographiclithography, Zernike's phase contrast method of observation of phasechanges, the Schlieren method of observation of phase changes, thecentral dark-background method of observation, spatial light modulators,acousto-optic cells, etc. In summary, it is intended that the scope ofthe present invention be defined by the claims appended hereto and theirequivalents.

What is claimed is:
 1. A method for metrology of additional-materialstructural deviations of a nominal periodic structure by comparison of ameasured diffraction spectrum from a target periodic structure with acalculated diffraction spectrum from a hypothetical deviated periodicstructure defined by applying said additional-material structuraldeviations to said nominal periodic structure, said hypotheticaldeviated periodic structure having a direction of periodicity x, adirection of essentially-infinite extension y orthogonal to saiddirection of periodicity x, and a normal direction z orthogonal to saiddirection of periodicity x and said direction of extension y,calculation of said calculated diffraction spectrum comprising the stepsof: defining a plurality of layers parallel to the x-y plane anddiscretizing an x-z plane cross-section of said periodic structure intoa plurality of stacked rectangular sections such that only two materialsfrom said nominal periodic structure are within each of said pluralityof layers, and only a single material lies along each line along thenormal direction z in each of said plurality of layers in saidhypothetical deviated periodic structure, and at least three materialsare within at least one of said plurality of layers in said hypotheticaldeviated periodic structure; performing a harmonic expansion of afunction of the permittivity ε along said direction of periodicity x foreach of said layers including said at least one of said plurality oflayers in said hypothetical deviated periodic structure having said atleast three materials therein; setting up Fourier space electromagneticequations in said each of said layers using said harmonic expansion ofsaid function of the permittivity ε for said each of said layers andFourier components of electric and magnetic fields; coupling saidFourier space electromagnetic equations based on boundary conditionsbetween said layers; and solving said coupling of said Fourier spaceelectromagnetic equations to provide said calculated diffractionspectrum.
 2. The method of claim 1 wherein said harmonic expansion ofsaid function of the permittivity ε along said direction of periodicityx for said at least one of said plurality of layers in said hypotheticaldeviated periodic structure having said at least three materials thereinis given by$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

for the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}\quad {\frac{j\quad n_{k}^{2}}{\quad 2\pi}\left\lbrack {\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right)}} \right\rbrack}}$

for the i^(th)-order harmonic component, where D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andx_(k−1), j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 3. The method of claim 1wherein said only two materials from said nominal periodic structurewithin at least a single one of said plurality of layers are a solid anda non-solid.
 4. The method of claim 1 wherein said periodic grating is asemiconductor grating with a critical dimension along said direction ofperiodicity x, and said additional-material structural deviations aredeviations along said normal direction z.
 5. The method of claim 4wherein said additional-material structural deviations along said normaldirection z result in an atmospheric region in what was a solid regionof said nominal periodic structure.
 6. The method of claim 1 whereinsaid periodic grating is a semiconductor grating with a criticaldimension along said direction of periodicity x, and saidadditional-material structural deviations are deviations along saiddirection of periodicity x due to a polymer deposit.
 7. The method ofclaim 1 wherein said periodic grating is a semiconductor grating andsaid additional-material structural deviations are purposefully includedto provide a structure having particular electronic characteristics. 8.The method of claim 1 wherein an initial one of said layers is anatmospheric region, and a final one of said layers is a substrate. 9.The method of claim 1 wherein said calculation of said calculateddiffraction spectrum is a rigorous coupled-wave calculation.
 10. Amethod of generating the diffracted reflectivity associated withdiffraction of electromagnetic radiation off a target periodic gratingto determine structural properties of the target periodic grating,including: dividing the target periodic grating into a plurality ofhypothetical layers, at least one of the hypothetical layers formedacross each of at least a first, second and third material, each of theat least first, second and third materials occurring along a directionof periodicity of the target periodic grating, each separatehypothetical layer having one of a plurality of possible combinations ofhypothetical values of properties for that hypothetical layer;generating sets of hypothetical layer data, each set of hypotheticallayer data corresponding to a separate one of the plurality ofhypothetical layers; and processing the generated sets of hypotheticallayer data to generate the diffracted reflectivity that would occur byreflecting electromagnetic radiation off the periodic grating.
 11. Themethod of claim 10 further including subdividing the hypothetical layersinto a plurality of slabs, each slab corresponding to the intersectionof one of the plurality of layers with one of at least the first, secondand third materials.
 12. The method of claim 11 wherein the step ofdividing the target periodic grating into a plurality of hypotheticallayers includes dividing the target periodic grating into a plurality ofhypothetical layers which are parallel to the direction of periodicityof the target periodic grating.
 13. The method of claim 10 wherein thestep of generating sets of hypothetical layer data includes expanding atleast one of either a function of a real space permittivity and afunction of a real space inverse permittivity of the hypothetical layersin a one-dimensional Fourier transformation along the direction ofperiodicity of the target periodic grating to provide harmoniccomponents of the at least one of either a function of a real spacepermittivity and a function of a real space inverse permittivity of thehypothetical layers.
 14. The method of claim 10 wherein the step ofgenerating sets of hypothetical layer data includes computing at leastone of: permittivity properties including a function of a permittivityε_(l)(x) of each of the hypothetical layers of the target periodicgrating, the harmonic components ε_(l,i) of the function of thepermittivity ε_(l)(x), and a permittivity harmonics matrix [E_(l)]; andinverse-permittivity properties including a function of aninverse-permittivity π_(l)(x) of each of the hypothetical layers of thetarget periodic grating, the harmonic components π_(l,i) of the functionof the inverse-permittivity π_(l)(x), and an inverse-permittivityharmonics matrix [P_(l)].
 15. The method of claim 14 wherein the step ofprocessing the generated sets of hypothetical layer data includes:computing a wave-vector matrix [A_(l)] by combining a series expansionof the electric field of each of the hypothetical layers of the targetperiodic grating with at least one of at least the permittivityharmonics matrix [E_(l)] and inverse-permittivity harmonics matrix[P_(l)]; and computing the ith entry w_(l,i,m) of the mth eigenvector ofthe wave-vector matrix [A_(l)] and the mth eigenvalue τ_(l,m) of thewave-vector matrix [A_(l)] to form an eigenvector matrix [W_(l)] and aroot-eigenvalue matrix [Q_(l)].
 16. The method of claim 10 wherein thestep of generating sets of hypothetical layer data includes expandingone of at least a function of a permittivity ε_(l)(x) and a function ofan inverse-permittivity π_(l)(x)=1/ε_(l)(x) of the at least one of thehypothetical layers formed across each of at least the first, second andthird materials of the target periodic grating in a one-dimensionalFourier transformation, the expansion performed along the direction ofperiodicity of the target periodic grating according to at least one of:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where$ɛ_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{n_{k}^{2}}{- {{j}2\pi}}\left\lbrack {\left( {{\cos \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\sin \left( {\frac{2i\quad \pi}{D}x_{k - 1}} \right)}} \right)}} \right\rbrack}}$and${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}}$

where$\left. {\pi_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{1}{- {{j}2}}\frac{1}{n_{k}^{2}}\left( {{\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} -} \right.}{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}} - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}}} \right)$

where D is the pitch of said hypothetical deviated periodic structure,n_(k) is the index of refraction of a material between materialboundaries at x_(k) and x_(k−1), j is the imaginary number defined asthe square root of −1, and there are r of said material boundarieswithin each period of said hypothetical deviated periodic structure. 17.The method of claim 10, wherein the step of processing the generatedsets of hypothetical layer data includes: constructing a matrix equationfrom the intermediate data corresponding to the hypothetical layers ofthe target periodic grating; and solving the constructed matrix equationto determine the diffracted reflectivity value R_(i) for each harmonicorder i.
 18. A method of generating the diffracted reflectivityassociated with diffraction of electromagnetic radiation off a targetperiodic grating to determine structural properties of the targetperiodic grating, including: dividing the target periodic grating into aplurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and third materialoccurring along a line parallel to a direction of periodicity of thetarget periodic grating; performing an harmonic expansion of a functionof the permittivity ε along the direction of periodicity of the targetperiod grating for each of the hypothetical layers including the atleast one of the plurality of layers formed across each of at least afirst, second and third material; setting up Fourier spaceelectromagnetic equations in each of the hypothetical layers using theharmonic expansion of the function of the permittivity ε for said eachof the hypothetical layers and Fourier components of electric andmagnetic fields; coupling the Fourier space electromagnetic equationsbased on boundary conditions between the layers; and solving thecoupling of the Fourier space electromagnetic equations to provide adiffracted reflectivity.
 19. The method of claim 18 further includingsubdividing at least one of the plurality of hypothetical layers into aplurality of hypothetical slabs, each hypothetical slab corresponding toan intersection of the at least one of the plurality of hypotheticallayers with one of at least the first, second and third materials. 20.The method of claim 19 wherein the step of subdividing at least one ofthe hypothetical layers into a plurality of hypothetical slabs includessubdividing the at least one hypothetical layer into a plurality ofhypothetical slabs such that only a single material lies along any lineperpendicular to the direction of periodicity of the target periodicgrating and normal to the target periodic grating.
 21. The method ofclaim 18 wherein the harmonic expansion of the function of thepermittivity along the direction of periodicity of the target periodicgrating for the at least one of the hypothetical layers formed acrosseach of at least the first, second and third material is given by:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where,$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}\quad {\frac{j\quad n_{k}^{2}}{{2}\quad \pi}\left\lbrack {\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right)}} \right\rbrack}}$

is the i^(th)-order harmonic component, l indicates the lth one of theplurality of hypothetical layers, D is the pitch of said hypotheticaldeviated periodic structure, n_(k) is the index of refraction of amaterial between material boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are r ofsaid material boundaries within each period of said hypotheticaldeviated periodic structure.
 22. A method of generating an expression ofthe permittivity of a target periodic grating having more than twomaterials in a periodic direction for use in an optical profilometryformalism for determining a diffracted reflectivity of the targetperiodic grating comprising: dividing the target periodic grating into aplurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and third materialoccurring along a line parallel to a direction of periodicity of thetarget periodic grating; subdividing at least one of the plurality ofhypothetical layers into a plurality of hypothetical slabs to generate aplurality of hypothetical boundaries, each of the plurality ofhypothetical boundaries corresponding to an intersection of at least oneof the plurality of hypothetical layers with one of at least the first,second and third materials; determining a permittivity function for eachof the plurality of hypothetical layers; completing a one-dimensionalFourier expansion of the permittivity function of each hypotheticallayer along the direction of periodicity of the target periodic gratingby summing the Fourier components over the plurality of hypotheticalboundaries to provide harmonic components of the at least onepermittivity function; and defining a permittivity harmonics matrixincluding the harmonic components of the Fourier expansion of thepermittivity function.
 23. The method of claim 22 wherein the step ofcompleting a one dimensional Fourier transform of a permittivityfunction includes completing the one dimensional Fourier transform ofthe permittivity function ε_(l)(x) according to:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$\left. {ɛ_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{n_{k}^{2}}{- {{j}2\pi}}\left\lbrack {\left( {{\cos \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} -} \right.{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}} - {j\left( {{\sin \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\sin \left( {\frac{2i\quad \pi}{D}x_{k - 1}} \right)}} \right)}}} \right\rbrack$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 24. The method of claim 23 wherein the permittivity harmonicsmatrix E_(l) is a (2o+1) x(2o+1) Toeplitz-form matrix having the form:$E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \cdots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \cdots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \cdots & ɛ_{l,{- {({{2o} - 2})}}} \\\cdots & \cdots & \cdots & \cdots & \cdots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \cdots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 25. The method of claim22 wherein the step of completing a one dimensional Fourier transform ofa permittivity function includes completing the one dimensional Fouriertransform of the permittivity function π_(l)(x) according to:${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}}$

where$\pi_{l,0} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{n_{k}^{2}}\frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,$\left. {\pi_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{1}{- {{j}2\pi}}\frac{1}{n_{k}^{2}}\left( {\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} -} \right.{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}} - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}}} \right)$

is the ith order component, where l indicates the lth one of theplurality of hypothetical layers, D is the pitch of the target periodicgrating, n_(k) is the index of refraction of a material between each ofthe plurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 26. The method of claim 25 wherein the permittivity harmonicsmatrix P_(l) is a (2o+1) x(2o+1) Toeplitz-form matrix having the form:$P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \cdots & \pi_{l,{2o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,1} & \cdots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \cdots & \pi_{l,{- {({{2o} - 2})}}} \\\cdots & \cdots & \cdots & \cdots & \cdots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \cdots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 27. A computer readablestorage medium containing computer executable code for generating thetheoretical diffracted reflectivity associated with diffraction ofelectromagnetic radiation off a target periodic grating to determinestructural properties of the target periodic grating by instructing acomputer to operate as follows: divide the target periodic grating intoa plurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and thirdmaterial, each of the at least first, second and third materialsoccurring along a direction of periodicity of the target periodicgrating, each separate hypothetical layer having one of a plurality ofpossible combinations of hypothetical values of properties for thathypothetical layer; generate sets of hypothetical layer data, each setof hypothetical layer data corresponding to a separate one of theplurality of hypothetical layers; and process the generated sets ofhypothetical layer data to generate the diffracted reflectivity thatwould occur by reflecting electromagnetic radiation off the periodicgrating.
 28. The computer readable storage medium of claim 27 whereinthe computer is further instructed to subdivide the hypothetical layersinto a plurality of slabs, each slab corresponding to the intersectionof one of the plurality of layers with one of at least the first, secondand third materials.
 29. The computer readable storage medium of claim28 wherein in dividing the target periodic grating into a plurality ofhypothetical layers the computer is instructed to divide the targetperiodic grating into a plurality of hypothetical layers which areparallel to the direction of periodicity of the target periodic grating.30. The computer readable medium of claim 27 wherein in generating setsof hypothetical layer data the computer is instructed to expand at leastone of either a function of a real space permittivity and a function ofa real space inverse permittivity of the hypothetical layers in aone-dimensional Fourier transformation along the direction ofperiodicity of the target periodic grating to provide harmoniccomponents of the at least one of either a function of a real spacepermittivity and a function of a real space inverse permittivity of thehypothetical layers.
 31. The computer readable storage medium of claim27 wherein in generating sets of hypothetical layer data the computer isinstructed to compute at least one of: permittivity properties includinga function of a permittivity ε_(l)(x) of each of the hypothetical layersof the target periodic grating, the harmonic components ε_(l,i) of thefunction of the permittivity ε_(l)(x), and a permittivity harmonicsmatrix [E_(l)]; and inverse-permittivity properties including a functionof an inverse-permittivity π_(l)(x) of each of the hypothetical layersof the target periodic grating, the harmonic components π_(l,i) of thefunction of the inverse-permittivity π_(l)(x), and aninverse-permittivity harmonics matrix [P_(l)].
 32. The computer readablemedium of claim 31 wherein in processing the generated sets ofhypothetical layer data the computer is instructed to: compute awave-vector matrix [A_(l)] by combining a series expansion of theelectric field of each of the hypothetical layers of the target periodicgrating with at least one of at least the permittivity harmonics matrix[E_(l)] and inverse-permittivity harmonics matrix [P_(l)]; and computethe ith entry w_(l,i,m) of the mth eigenvector of the wave-vector matrix[A_(l)] and the mth eigenvalue τ_(l,m) of the wave-vector matrix [A_(l)]to form an eigenvector matrix [W_(l)] and a root-eigenvalue matrix[Q_(l].)
 33. The computer readable medium of claim 27 wherein ingenerating sets of hypothetical layer data the computer is instructed toexpand one of at least a permittivity ε_(l)(x) and aninverse-permittivity π_(l)(x)=1/ε_(l)(x) of the at least one of thehypothetical layers formed across each of at least the first, second andthird materials of the target periodic grating in a one-dimensionalFourier transformation, the expansion performed along the direction ofperiodicity of the target periodic grating according to at least one of:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where$ɛ_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{n_{k}^{2}}{- {{j}2\pi}}\left\lbrack {\left( {{\cos \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2i\quad \pi}{D}x_{k}} \right)} - {\sin \left( {\frac{2i\quad \pi}{D}x_{k - 1}} \right)}} \right)}} \right\rbrack}}$and${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}}$

where$\left. {\pi_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{1}{- {{j}2}}\frac{1}{n_{k}^{2}}\left( {{\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} -} \right.}{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}} - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}}} \right)$

where l indicates the lth one of the plurality of hypothetical layers, Dis the pitch of said hypothetical deviated periodic structure, n_(k) isthe index of refraction of a material between material boundaries atx_(k) and x_(k−1), j is the imaginary number defined as the square rootof −1and there are r of said material boundaries within each period ofsaid hypothetical deviated periodic structure.
 34. The computer readablemedium of claim 27, wherein in processing the generated sets ofhypothetical layer data the computer is configured to: construct amatrix equation from the intermediate data corresponding to thehypothetical layers of the target periodic grating; and solve theconstructed matrix equation to determine the diffracted reflectivityvalue R_(i) for each harmonic order i.
 35. A computer readable storagemedium containing computer executable code for generating the diffractedreflectivity associated with diffraction of electromagnetic radiationoff a target periodic grating to determine structural properties of thetarget periodic grating by instructing a computer to operate as follows:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; perform anharmonic expansion of a function of the permittivity ε_(l)(x) along thedirection of periodicity of the target period grating for each of thehypothetical layers including the at least one of the plurality oflayers formed across each of at least a first, second and thirdmaterial; set up Fourier space electromagnetic equations in each of thehypothetical layers using the harmonic expansion of the function of thepermittivity ε_(l)(x) for said each of the hypothetical layers andFourier components of electric and magnetic fields; couple the Fourierspace electromagnetic equations based on boundary conditions between thelayers; and solve the coupling of the Fourier space electromagneticequations to provide a diffracted reflectivity.
 36. The computerreadable storage medium of claim 35 wherein the computer is furtherinstructed to subdivide at least one of the plurality of hypotheticallayers into a plurality of hypothetical slabs, each hypothetical slabcorresponding to an intersection of the at least one of the plurality ofhypothetical layers with one of at least the first, second and thirdmaterials.
 37. The computer readable storage medium of claim 36 whereinin subdividing the at least one of the hypothetical layers into aplurality of hypothetical slabs the computer executable code instructsthe computer to subdivide the at least one hypothetical layer into aplurality of hypothetical slabs such that only a single material liesalong any line perpendicular to the direction of periodicity of thetarget periodic grating and normal to the target periodic grating. 38.The computer readable storage medium of claim 37 wherein in performingan harmonic expansion of a function of the permittivity ε_(l)(x) thecomputer code instructs the computer to perform the harmonic expansionsuch that:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where:$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

for the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}\quad {\frac{j\quad n_{k}^{2}}{2\pi}\left\lbrack {\left( {{\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right) - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)}} \right)}} \right\rbrack}}$

for the i^(th)-order harmonic component, where l indicates the lth oneof the plurality of hypothetical layers, D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andx_(k−1), j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 39. A computer readablestorage medium containing computer executable code for generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction for use in an opticalprofilometry formalism for determining a diffracted reflectivity of thetarget periodic grating by instructing a computer to operate as follows:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; subdivideat least one of the plurality of hypothetical layers into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries,each of the plurality of hypothetical boundaries corresponding to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials; determine apermittivity function for each of the plurality of hypothetical layers;complete a one-dimensional Fourier expansion of the permittivityfunction of each hypothetical layer along the direction of periodicityof the target periodic grating by summing the Fourier components overthe plurality of hypothetical boundaries to provide harmonic componentsof the at least one permittivity function; and define a permittivityharmonics matrix including the harmonic components of the Fourierexpansion of the permittivity function.
 40. The computer readablestorage medium of claim 39 wherein in completing a one dimensionalFourier transform of a permittivity function the computer executablecode instructs the computer to complete the one dimensional Fouriertransform of the permittivity function ε_(l)(x) according to:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}{\exp \left( {j\frac{2\pi \quad i}{D}x} \right)}}}$

where$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$\left. {ɛ_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{\quad n_{k}^{2}}{- {j2\pi}}\left\lbrack {\left( {\cos \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} \right. - {\cos \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}} - {j\left( {{\sin \left( {\frac{2\pi \quad i}{D}x_{k}} \right)} - {\sin \left( {\frac{2\pi \quad i}{D}x_{k - 1}} \right)}} \right)}}} \right\rbrack$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 41. The computer readable storage medium of claim 40 wherein indefining a permittivity harmonics matrix E_(l) the computer executablecode instructs the computer to generate a (2o+1)×(2o+1) Toeplitz-formmatrix having the form: $E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 42. The computerreadable storage medium of claim 39 wherein in completing a onedimensional Fourier transform of a permittivity function the computerexecutable code instructs the computer to complete the one dimensionalFourier transform of the permittivity function π_(l)(x) according to:${\pi_{l}\quad (x)} = {\frac{1}{ɛ_{l}\quad (x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}}$

where$\pi_{l,0} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{n_{k}^{2}}\quad \frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,$\pi_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{{- {ji2}}\quad \pi}\quad \frac{1}{n_{k}^{2}}\quad \left( {\left( {{\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right) - {j\quad \left( {{\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right)}} \right)}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 43. The computer readable storage medium of claim 42 wherein indefining a permittivity harmonics matrix P_(l) the computer executablecode instructs the computer to generate a (2o+1)×(2o+1) Toeplitz-formmatrix having the form: $P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 44. A system forgenerating a theoretical diffracted reflectivity associated withdiffraction of electromagnetic radiation off a target periodic gratingto determine structural properties of the target periodic grating,including a computer processor configured to: divide the target periodicgrating into a plurality of hypothetical layers, at least one of thehypothetical layers formed across each of at least a first, second andthird material, each of the at least first, second and third materialsoccurring along a direction of periodicity of the target periodicgrating, each separate hypothetical layer having one of a plurality ofpossible combinations of hypothetical values of properties for thathypothetical layer; generate sets of hypothetical layer data, each setof hypothetical layer data corresponding to a separate one of theplurality of hypothetical layers; and process the generated sets ofhypothetical layer data to generate the diffracted reflectivity thatwould occur by reflecting electromagnetic radiation off the periodicgrating.
 45. The system of claim 44 wherein the computer processor isfurther configured to subdivide the hypothetical layers into a pluralityof slabs, each slab corresponding to the intersection of one of theplurality of layers with one of at least the first, second and thirdmaterials.
 46. The system of claim 45 wherein in dividing the targetperiodic grating into a plurality of hypothetical layers the computerprocessor is further configured to divide the target periodic gratinginto a plurality of hypothetical layers which are parallel to thedirection of periodicity of the target periodic grating.
 47. The systemof claim 44 wherein in generating sets of hypothetical layer data thecomputer processor is configured to expand at least one of either afunction of a real space permittivity and a.function of a real spaceinverse permittivity of the hypothetical layers in a one-dimensionalFourier transformation along the direction of periodicity of the targetperiodic grating to provide harmonic components of the at least one ofeither a function of a real space permittivity and a function of a realspace inverse permittivity of the hypothetical layers.
 48. The system ofclaim 44 wherein in generating sets of hypothetical layer data thecomputer processor is configured to compute at least one of:permittivity properties including a function of a permittivity ε_(l)(x)of each of the hypothetical layers of the target periodic grating, theharmonic components ε_(l,i) of the function of the permittivityε_(l)(x), and a permittivity harmonics matrix [E_(l)]; andinverse-permittivity properties including a function of aninverse-permittivity π_(l)(x) of each of the hypothetical layers of thetarget periodic grating, the harmonic components π_(l,i) of the functionof the inverse-permittivity π_(l),(x), and an inverse-permittivityharmonics matrix [P_(l)].
 49. The system of claim 48 wherein inprocessing the generated sets of hypothetical layer data the computerprocessor is configured to: compute a wave-vector matrix [A_(l)] bycombining a series expansion of the electric field of each of thehypothetical layers of the target periodic grating with at least one ofat least the permittivity harmonics matrix [E_(l)] andinverse-permittivity harmonics matrix [P_(l)]. compute the ith entryw_(l,i,m) of the mth eigenvector of the wave-vector matrix [A_(l)] andthe mth eigenvalue τ_(l,m) of the wave-vector matrix [A_(l)] to form aneigenvector matrix [W_(l)] and a root-eigenvalue matrix [Q_(l)].
 50. Thesystem of claim 44 wherein in generating sets of hypothetical layer datathe computer processor is configured to expand one of at least afunction of a permittivity ε_(l)(x) and a function of an inversepermittivity π_(l)(x)=1/ε_(l)(x) of the at least one of the hypotheticallayers formed across each of at least the first, second and thirdmaterials of the target periodic grating in a one-dimensional Fouriertransformation, the expansion performed along the direction ofperiodicity of the target periodic grating according to at least one of:${ɛ_{l}\quad (x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}$

where$\left. \left. {ɛ_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}\left\lbrack \left( {{\cos \quad \left( {\frac{2i\quad \pi}{D}\quad x_{k}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right) \right.}} - {j\quad\left( \sin \quad \left( {\frac{2i\quad \pi}{D}\quad x_{k}} \right) \right.} - {\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}}} \right) \right\rbrack$and${\pi_{l}\quad (x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}}$

where$\pi_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{{- {ji2}}\quad \pi}\quad \frac{1}{n_{k}^{2}}\quad \left( {\left( {{\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right) - {j\quad \left( {{\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\sin \quad \left( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} \right)}} \right)}} \right)}}$

where l indicates the lth one of the plurality of hypothetical layers, Dis the pitch of said hypothetical deviated periodic structure, n_(k) isthe index of refraction of a material between material boundaries atx_(k) and x_(k−1), j is the imaginary number defined as the square rootof −1, and there are r of said material boundaries within each period ofsaid hypothetical deviated periodic structure.
 51. The system of claim44, wherein in processing the generated sets of hypothetical layer datathe computer processor is configured to: construct a matrix equationfrom the intermediate data corresponding to the hypothetical layers ofthe target periodic grating; and solve the constructed matrix equationto determine the diffracted reflectivity value R_(i) for each harmonicorder i.
 52. A system of generating the diffracted reflectivityassociated with diffraction of electromagnetic radiation off a targetperiodic grating to determine structural properties of the targetperiodic grating, including a computer microprocessor configured to:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; perform anharmonic expansion of a function of the permittivity ε along thedirection of periodicity of the target period grating for each of thehypothetical layers including the at least one of the plurality oflayers formed across each of at least a first, second and thirdmaterial; set up Fourier space electromagnetic equations in each of thehypothetical layers using the harmonic expansion of the function of thepermittivity ε for said each of the hypothetical layers and Fouriercomponents of electric and magnetic fields; couple the Fourier spaceelectromagnetic equations based on boundary conditions between thelayers; and solve the coupling of the Fourier space electromagneticequations to provide a diffracted reflectivity.
 53. The system of claim52 wherein the computer processor is further configured to subdivide atleast one of the plurality of hypothetical layers into a plurality ofhypothetical slabs, each hypothetical slab corresponding to anintersection of the at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials.
 54. Thesystem of claim 53 wherein in subdividing at least one of thehypothetical layers into a plurality of hypothetical slabs, the computerprocessor is configured to subdivide the at least one hypothetical layerinto a plurality of hypothetical slabs such that only a single materiallies along any line perpendicular to the direction of periodicity of thetarget periodic grating and normal to the target periodic grating. 55.The system of claim 52 wherein in performing an harmonic expansion of afunction of the permittivity the computer processor is configured togenerate the harmonic expansion of the function of the permittivityalong the direction of periodicity of the target periodic grating forthe at least one of the hypothetical layers formed across each of atleast the first, second and third material such that:${ɛ_{l}\quad (x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}$

where,$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\quad \frac{x_{k - 1} - x_{k}}{D}}}$

for the zeroth-order component, and${ɛ_{i} = {\sum\limits_{k = 1}^{r}\quad {\frac{{jn}_{k}^{2}}{{i2}\quad \pi}\left\lbrack {\left( {{\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)}} \right) - {j\quad \left( {{\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)} - {\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)}} \right)}} \right\rbrack}}}\quad$

for the i^(th)-order harmonic component, and where l indicates the lthone of the plurality of hypothetical layers, D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andx_(k−1), j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 56. A system of generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction for use in an opticalprofilometry formalism for determining a diffracted reflectivity of thetarget periodic grating including a computer processor configured to:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; subdivideat least one of the plurality of hypothetical layers into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries,each of the plurality of hypothetical boundaries corresponding to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials; determine apermittivity function for each of the plurality of hypothetical layers;complete a one-dimensional Fourier expansion of the permittivityfunction of each hypothetical layer along the direction of periodicityof the target periodic grating by summing the Fourier components overthe plurality of hypothetical boundaries to provide harmonic componentsof the at least one permittivity function; and define a permittivityharmonics matrix including the harmonic components of the Fourierexpansion of the permittivity function.
 57. The system of claim 56wherein in completing a one dimensional Fourier transform of apermittivity function the computer processor is configured to complete aone dimensional Fourier transform of the permittivity function ε_(l)(x)according to:${ɛ_{l}\quad (x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {ɛ_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}$

where$ɛ_{0} = {\sum\limits_{k = 1}^{r}\quad {n_{k}^{2}\quad \frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$ɛ_{l,i} = {{\sum\limits_{k = 1}^{r}\quad {\frac{n_{k}^{2}}{{ji2}\quad \pi}\left\lbrack \quad {\left( {{\cos \quad \left( {\frac{2i\quad \pi}{D}\quad x_{k}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right) - {j\quad \left( {{\sin \quad \left( {\frac{2i\quad \pi}{D}\quad x_{k}} \right)} - {\sin \quad \left( {\frac{2i\quad \pi}{D}\quad x_{k - 1}} \right)}} \right)}} \right\rbrack}}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 58. The system of claim 57 wherein in defining the permittivityharmonics matrix E_(l) the computer processor constructs a (2o+1)×(2o+1)Toeplitz-form matrix having the form: $E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 59. The system of claim56 wherein in completing a one dimensional Fourier transform of apermittivity function the computer processor completes a one dimensionalFourier transform of the permittivity function π_(l)(x) according to:${\pi_{l}\quad (x)} = {\frac{1}{ɛ_{l}\quad (x)} = {\sum\limits_{i = {- \infty}}^{\infty}\quad {\pi_{l,i}\quad \exp \quad \left( {j\quad \frac{2\quad \pi \quad i}{D}\quad x} \right)}}}$

where$\pi_{l,0} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{n_{k}^{2}}\quad \frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,${\pi_{l,i} = {\sum\limits_{k = 1}^{r}\quad {\frac{1}{{- {ji2}}\quad \pi}\quad \frac{1}{n_{k}^{2}}\quad \left( {\left( {{\cos \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\cos \quad \left( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} \right)}} \right) - {j\quad \left( {{\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k}} \right)} - {\sin \quad \left( {\frac{2\quad \pi \quad i}{D}\quad x_{k - 1}} \right)}} \right)}} \right)}}}\quad$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 60. The system of claim 59 wherein in defining the permittivityharmonics matrix P_(l) the computer processor constructs a (2o+1)×(2o+1)Toeplitz-form matrix having the form: $P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.